make 'complete euler cubes' of order 5 for the pan...

1

Part 4 : "New Advanced Study of Magic Squares and Cubes" Chapter 4 : Commentary Articles No.2 by Kanji Setsuda: "Various Arts and Tools for Studying Magic Squares" Section 15-2: Make 'Complete Euler Cubes' of Order 5 for the Pan-triagonal Type 0. What I like to do in this Section: I would like to challenge next to make Pan-triagonal Magic Cubes of Order 5. Since it would be too hard for us to make any ordinary type of this object, I would like to build the 'Complete Euler Type' instead, under the Positional Number System of Base 5 and count all the standard solutions through. 1. What the 'Pan-triagonal' Magic Cubes of Order 5 mean: At first I like to explain a few things about what it means by the 'Pan-triagonal' Magic Cubes of Order 5 with a set of diagrams below. Each of them illustrates the unique 'Pan-magic Shift' of our three-dimensional pan-magic cubes of order 5. Suppose Fig.1 for our Basic Diagram representing the Basic Form of MC555. Watch the next Fig.2 carefully. What is the difference between Fig.1 and Fig.2? Fig.1: [Basic Diagram] Fig.2 1----------26----------51----------76---------101 2----------27----------52----------77---------102 ¦ 2 27 52 77 102 ¦ 3 28 53 78 103

6 3 31 28 56 53 81 78 106 103 7 4 32 29 57 54 82 79 107 104

¦ 7 4 32 29 57 54 82 79 107 104 ¦ 8 5 33 30 58 55 83 80 108 105 11 8 5--36--33--30--61--58--55--86--83--80-111-108-105 12 9 1--37--34--26--62--59--51--87--84--76-112-109-101

¦12 9 ¦ 37 34 62 59 87 84 112 109 ¦ ¦13 10 ¦ 38 35 63 60 88 85 113 110 ¦

16 13 10 41 38 35 66 63 60 91 88 85 116 113 110 17 14 6 42 39 31 67 64 56 92 89 81 117 114 106 ¦17 14 ¦ 42 39 67 64 92 89 117 114 ¦ ¦18 15 ¦ 43 40 68 65 93 90 118 115 ¦

21--18--15--46--43--40--71--68--65--96--93--90-121 118 115 22--19--11--47--44--36--72--69--61--97--94--86-122 119 111

22 19 ¦ 47 44 72 69 97 94 122 119 ¦ 23 20 ¦ 48 45 73 70 98 95 123 120 ¦ 23 20 48 45 73 70 98 95 123 120 24 16 49 41 74 66 99 91 124 116

24 ¦ 49 74 99 124 ¦ 25 ¦ 50 75 100 125 ¦

25----------50----------75---------100---------125 21----------46----------71----------96---------121

n1+n32+n63+n94+n125=C ...pt11; ¦ n2+n33+n64+n95+n121=C ...pt21;




Fig.2 is transformed from Fig.1, moving the vertical plane with 25 numbers at the back side far beyond the front end. This tells us about what the Pan-magic Shift does. I want you to check if each sum of rows, columns and pillars is kept equal to the previous value, say, the constant sum. How about the new four Primary Triagonals? What do we have to do, if we want the new triagonals to be always equal to the constant sum? How about defining the conditions pt21-24 above as true before all? Let's do that and suppose all these four to be the new 'Pan-triagonals'.

3----------28----------53----------78---------103 4----------29----------54----------79---------104

¦ 4 29 54 79 104 ¦ 5 30 55 80 105 8 5 33 30 58 55 83 80 108 105 9 1 34 26 59 51 84 76 109 101

¦ 9 1 34 26 59 51 84 76 109 101 ¦10 2 35 27 60 52 85 77 110 102

13 10 2--38--35--27--63--60--52--88--85--77-113-110-102 14 6 3--39--31--28--64--56--53--89--81--78-114-106-103 ¦14 6 ¦ 39 31 64 56 89 81 114 106 ¦ ¦15 7 ¦ 40 32 65 57 90 82 115 107 ¦

18 15 7 43 40 32 68 65 57 93 90 82 118 115 107 19 11 8 44 36 33 69 61 58 94 86 83 119 111 108

¦19 11 ¦ 44 36 69 61 94 86 119 111 ¦ ¦20 12 ¦ 45 37 70 62 95 87 120 112 ¦ 23--20--12--48--45--37--73--70--62--98--95--87-123 120 112 24--16--13--49--41--38--74--66--63--99--91--88-124 116 113

24 16 ¦ 49 41 74 66 99 91 124 116 ¦ 25 17 ¦ 50 42 75 67 100 92 125 117 ¦

25 17 50 42 75 67 100 92 125 117 21 18 46 43 71 68 96 93 121 118 21 ¦ 46 71 96 121 ¦ 22 ¦ 47 72 97 122 ¦

22----------47----------72----------97---------122 23----------48----------73----------98---------123

2



n23+n44+n65+n81+n102=C ...pt33; ¦ n24+n45+n61+n82+n103=C ...pt43; n22+n41+n65+n84+n103=C ...pt34; ¦ n23+n42+n61+n85+n104=C ...pt44;

5----------30----------55----------80---------105 6----------31----------56----------81---------106 ¦ 1 26 51 76 101 ¦ 7 32 57 82 107

10 2 35 27 60 52 85 77 110 102 11 8 36 33 61 58 86 83 111 108

¦ 6 3 31 28 56 53 81 78 106 103 ¦12 9 37 34 62 59 87 84 112 109 15 7 4--40--32--29--65--57--54--90--82--79-115-107-104 16 13 10--41--38--35--66--63--60--91--88--85-116-113-110

¦11 8 ¦ 36 33 61 58 86 83 111 108 ¦ ¦17 14 ¦ 42 39 67 64 92 89 117 114 ¦

20 12 9 45 37 34 70 62 59 95 87 84 120 112 109 21 18 15 46 43 40 71 68 65 96 93 90 121 118 115 ¦16 13 ¦ 41 38 66 63 91 88 116 113 ¦ ¦22 19 ¦ 47 44 72 69 97 94 122 119 ¦

25--17--14--50--42--39--75--67--64-100--92--89-125 117 114 1--23--20--26--48--45--51--73--70--76--98--95-101 123 120

21 18 ¦ 46 43 71 68 96 93 121 118 ¦ 2 24 ¦ 27 49 52 74 77 99 102 124 ¦ 22 19 47 44 72 69 97 94 122 119 3 25 28 50 53 75 78 100 103 125

23 ¦ 48 73 98 123 ¦ 4 ¦ 29 54 79 104 ¦

24----------49----------74----------99---------124 5----------30----------55----------80---------105




Successive diagrams Fig.3, 4 and 5 are transformed from the former one, repeating the continuous shift of the vertical plane on the back side up to before the front. If you want every result to be always pan-magic, you should presuppose all the new conditions above to be true for the Pan-triagonals at the first definition stage. Fig.6 illustrates a new style of Pan-magic Shift. Fig.6 is transformed from the basic Fig.1, moving the horizontal plane with 25 numbers on the top down to the bottom end. Successive diagrams Fig.7-10 are transformed from the former one, repeating the first type of shift with the plane moving from the back side far to the front end. 7----------32----------57----------82---------107 8----------33----------58----------83---------108

¦ 8 33 58 83 108 ¦ 9 34 59 84 109

12 9 37 34 62 59 87 84 112 109 13 10 38 35 63 60 88 85 113 110 ¦13 10 38 35 63 60 88 85 113 110 ¦14 6 39 31 64 56 89 81 114 106

17 14 6--42--39--31--67--64--56--92--89--81-117-114-106 18 15 7--43--40--32--68--65--57--93--90--82-118-115-107

¦18 15 ¦ 43 40 68 65 93 90 118 115 ¦ ¦19 11 ¦ 44 36 69 61 94 86 119 111 ¦ 22 19 11 47 44 36 72 69 61 97 94 86 122 119 111 23 20 12 48 45 37 73 70 62 98 95 87 123 120 112

¦23 20 ¦ 48 45 73 70 98 95 123 120 ¦ ¦24 16 ¦ 49 41 74 66 99 91 124 116 ¦

2--24--16--27--49--41--52--74--66--77--99--91-102 124 116 3--25--17--28--50--42--53--75--67--78-100--92-103 125 117 3 25 ¦ 28 50 53 75 78 100 103 125 ¦ 4 21 ¦ 29 46 54 71 79 96 104 121 ¦

4 21 29 46 54 71 79 96 104 121 5 22 30 47 55 72 80 97 105 122

5 ¦ 30 55 80 105 ¦ 1 ¦ 26 51 76 101 ¦ 1----------26----------51----------76---------101 2----------27----------52----------77---------102




9----------34----------59----------84---------109 10----------35----------60----------85---------110 ¦10 35 60 85 110 ¦ 6 31 56 81 106

14 6 39 31 64 56 89 81 114 106 15 7 40 32 65 57 90 82 115 107

¦15 7 40 32 65 57 90 82 115 107 ¦11 8 36 33 61 58 86 83 111 108 19 11 8--44--36--33--69--61--58--94--86--83-119-111-108 20 12 9--45--37--34--70--62--59--95--87--84-120-112-109

¦20 12 ¦ 45 37 70 62 95 87 120 112 ¦ ¦16 13 ¦ 41 38 66 63 91 88 116 113 ¦

24 16 13 49 41 38 74 66 63 99 91 88 124 116 113 25 17 14 50 42 39 75 67 64 100 92 89 125 117 114 ¦25 17 ¦ 50 42 75 67 100 92 125 117 ¦ ¦21 18 ¦ 46 43 71 68 96 93 121 118 ¦

4--21--18--29--46--43--54--71--68--79--96--93-104 121 118 5--22--19--30--47--44--55--72--69--80--97--94-105 122 119

5 22 ¦ 30 47 55 72 80 97 105 122 ¦ 1 23 ¦ 26 48 51 73 76 98 101 123 ¦ 1 23 26 48 51 73 76 98 101 123 2 24 27 49 52 74 77 99 102 124

2 ¦ 27 52 77 102 ¦ 3 ¦ 28 53 78 103 ¦

3----------28----------53----------78---------103 4----------29----------54----------79---------104




3

If you want every result to be always pan-magic, you should add all those new conditions above to the first definition list for the Pan-triagonals. Fig.11 is transformed from the Fig.6, moving the horizontal plane with 25 numbers on the top down to beneath the bottom. Successive Fig.12-15 are transformed from the former one by the first type, repeating the continuous shift of the plane from the back side to the front end. 11----------36----------61----------86---------111 12----------37----------62----------87---------112

¦12 37 62 87 112 ¦13 38 63 88 113

16 13 41 38 66 63 91 88 116 113 17 14 42 39 67 64 92 89 117 114 ¦17 14 42 39 67 64 92 89 117 114 ¦18 15 43 40 68 65 93 90 118 115

21 18 15--46--43--40--71--68--65--96--93--90-121-118-115 22 19 11--47--44--36--72--69--61--97--94--86-122-119-111

¦22 19 ¦ 47 44 72 69 97 94 122 119 ¦ ¦23 20 ¦ 48 45 73 70 98 95 123 120 ¦ 1 23 20 26 48 45 51 73 70 76 98 95 101 123 120 2 24 16 27 49 41 52 74 66 77 99 91 102 124 116

¦ 2 24 ¦ 27 49 52 74 77 99 102 124 ¦ ¦ 3 25 ¦ 28 50 53 75 78 100 103 125 ¦

6-- 3--25--31--28--50--56--53--75--81--78-100-106 103 125 7-- 4--21--32--29--46--57--54--71--82--79--96-107 104 121 7 4 ¦ 32 29 57 54 82 79 107 104 ¦ 8 5 ¦ 33 30 58 55 83 80 108 105 ¦

8 5 33 30 58 55 83 80 108 105 9 1 34 26 59 51 84 76 109 101

9 ¦ 34 59 84 109 ¦ 10 ¦ 35 60 85 110 ¦ 10----------35----------60----------85---------110 6----------31----------56----------81---------106




13----------38----------63----------88---------113 14----------39----------64----------89---------114

¦14 39 64 89 114 ¦15 40 65 90 115

18 15 43 40 68 65 93 90 118 115 19 11 44 36 69 61 94 86 119 111 ¦19 11 44 36 69 61 94 86 119 111 ¦20 12 45 37 70 62 95 87 120 112

23 20 12--48--45--37--73--70--62--98--95--87-123-120-112 24 16 13--49--41--38--74--66--63--99--91--88-124-116-113

¦24 16 ¦ 49 41 74 66 99 91 124 116 ¦ ¦25 17 ¦ 50 42 75 67 100 92 125 117 ¦ 3 25 17 28 50 42 53 75 67 78 100 92 103 125 117 4 21 18 29 46 43 54 71 68 79 96 93 104 121 118

¦ 4 21 ¦ 29 46 54 71 79 96 104 121 ¦ ¦ 5 22 ¦ 30 47 55 72 80 97 105 122 ¦

8-- 5--22--33--30--47--58--55--72--83--80--97-108 105 122 9-- 1--23--34--26--48--59--51--73--84--76--98-109 101 123 9 1 ¦ 34 26 59 51 84 76 109 101 ¦ 10 2 ¦ 35 27 60 52 85 77 110 102 ¦

10 2 35 27 60 52 85 77 110 102 6 3 31 28 56 53 81 78 106 103

6 ¦ 31 56 81 106 ¦ 7 ¦ 32 57 82 107 ¦ 7----------32----------57----------82---------107 8----------33----------58----------83---------108




15----------40----------65----------90---------115 16----------41----------66----------91---------116 ¦11 36 61 86 111 ¦17 42 67 92 117

20 12 45 37 70 62 95 87 120 112 21 18 46 43 71 68 96 93 121 118

¦16 13 41 38 66 63 91 88 116 113 ¦22 19 47 44 72 69 97 94 122 119 25 17 14--50--42--39--75--67--64-100--92--89-125-117-114 1 23 20--26--48--45--51--73--70--76--98--95-101-123-120

¦21 18 ¦ 46 43 71 68 96 93 121 118 ¦ ¦ 2 24 ¦ 27 49 52 74 77 99 102 124 ¦

5 22 19 30 47 44 55 72 69 80 97 94 105 122 119 6 3 25 31 28 50 56 53 75 81 78 100 106 103 125 ¦ 1 23 ¦ 26 48 51 73 76 98 101 123 ¦ ¦ 7 4 ¦ 32 29 57 54 82 79 107 104 ¦

10-- 2--24--35--27--49--60--52--74--85--77--99-110 102 124 11-- 8-- 5--36--33--30--61--58--55--86--83--80-111 108 105

6 3 ¦ 31 28 56 53 81 78 106 103 ¦ 12 9 ¦ 37 34 62 59 87 84 112 109 ¦ 7 4 32 29 57 54 82 79 107 104 13 10 38 35 63 60 88 85 113 110

8 ¦ 33 58 83 108 ¦ 14 ¦ 39 64 89 114 ¦

9----------34----------59----------84---------109 15----------40----------65----------90---------115




. . . . .

If you want every result to be always pan-magic, you should also add all those new conditions above to the first definition list for the Pan-triagonals. Fig.16 and 21 is transformed from the Fig.6 and 11, moving the horizontal plane on the top far to the bottom end. Successive Fig.22-25 are transformed from the former one by the first type, repeating plane-shift from the back side to the front end.

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21----------46----------71----------96---------121 22----------47----------72----------97---------122

¦22 47 72 97 122 ¦23 48 73 98 123

1 23 26 48 51 73 76 98 101 123 2 24 27 49 52 74 77 99 102 124 ¦ 2 24 27 49 52 74 77 99 102 124 ¦ 3 25 28 50 53 75 78 100 103 125

6 3 25--31--28--50--56--53--75--81--78-100-106-103-125 7 4 21--32--29--46--57--54--71--82--79--96-107-104-121

¦ 7 4 ¦ 32 29 57 54 82 79 107 104 ¦ ¦ 8 5 ¦ 33 30 58 55 83 80 108 105 ¦ 11 8 5 36 33 30 61 58 55 86 83 80 111 108 105 12 9 1 37 34 26 62 59 51 87 84 76 112 109 101

¦12 9 ¦ 37 34 62 59 87 84 112 109 ¦ ¦13 10 ¦ 38 35 63 60 88 85 113 110 ¦

16--13--10--41--38--35--66--63--60--91--88--85-116 113 110 17--14-- 6--42--39--31--67--64--56--92--89--81-117 114 106 17 14 ¦ 42 39 67 64 92 89 117 114 ¦ 18 15 ¦ 43 40 68 65 93 90 118 115 ¦

18 15 43 40 68 65 93 90 118 115 19 11 44 36 69 61 94 86 119 111

19 ¦ 44 69 94 119 ¦ 20 ¦ 45 70 95 120 ¦ 20----------45----------70----------95---------120 16----------41----------66----------91---------116



n20+n39+n58+n77+n121=C ...pt214; ¦ n16+n40+n59+n78+n122=C ...pt224; 23----------48----------73----------98---------123 24----------49----------74----------99---------124

¦24 49 74 99 124 ¦25 50 75 100 125 3 25 28 50 53 75 78 100 103 125 4 21 29 46 54 71 79 96 104 121

¦ 4 21 29 46 54 71 79 96 104 121 ¦ 5 22 30 47 55 72 80 97 105 122

8 5 22--33--30--47--58--55--72--83--80--97-108-105-122 9 1 23--34--26--48--59--51--73--84--76--98-109-101-123 ¦ 9 1 ¦ 34 26 59 51 84 76 109 101 ¦ ¦10 2 ¦ 35 27 60 52 85 77 110 102 ¦

13 10 2 38 35 27 63 60 52 88 85 77 113 110 102 14 6 3 39 31 28 64 56 53 89 81 78 114 106 103

¦14 6 ¦ 39 31 64 56 89 81 114 106 ¦ ¦15 7 ¦ 40 32 65 57 90 82 115 107 ¦ 18--15-- 7--43--40--32--68--65--57--93--90--82-118 115 107 19--11-- 8--44--36--33--69--61--58--94--86--83-119 111 108

19 11 ¦ 44 36 69 61 94 86 119 111 ¦ 20 12 ¦ 45 37 70 62 95 87 120 112 ¦

20 12 45 37 70 62 95 87 120 112 16 13 41 38 66 63 91 88 116 113 16 ¦ 41 66 91 116 ¦ 17 ¦ 42 67 92 117 ¦

17----------42----------67----------92---------117 18----------43----------68----------93---------118



n17+n36+n60+n79+n123=C ...pt234; ¦ n18+n37+n56+n80+n124=C ...pt244; 25----------50----------75---------100---------125 26----------51----------76---------101---------- 1

¦21 46 71 96 121 ¦27 52 77 102 ¦ 2 5 22 30 47 55 72 80 97 105 122 31 28 56 53 81 78 106 103 6 3

¦ 1 23 26 48 51 73 76 98 101 123 ¦32 29 57 54 82 79 107 104 ¦ 7 4

10 2 24--35--27--49--60--52--74--85--77--99-110-102-124 36 33 30--61--58--55--86--83--80-111-108-105--11-- 8-- 5 ¦ 6 3 ¦ 31 28 56 53 81 78 106 103 ¦ ¦37 34 ¦ 62 59 87 84 112 109 ¦12 9 ¦

15 7 4 40 32 29 65 57 54 90 82 79 115 107 104 41 38 35 66 63 60 91 88 85 116 113 110 16 13 10

¦11 8 ¦ 36 33 61 58 86 83 111 108 ¦ ¦42 39 ¦ 67 64 92 89 117 114 ¦17 14 ¦ 20--12-- 9--45--37--34--70--62--59--95--87--84-120 112 109 46--43--40--71--68--65--96--93--90-121-118-115--21 18 15

16 13 ¦ 41 38 66 63 91 88 116 113 ¦ 47 44 ¦ 72 69 97 94 122 119 22 19 ¦

17 14 42 39 67 64 92 89 117 114 48 45 73 70 98 95 123 120 23 20 18 ¦ 43 68 93 118 ¦ 49 ¦ 74 99 124 24 ¦

19----------44----------69----------94---------119 50----------75---------100---------125----------25

n25+n26+n57+n88+n119=C ...pt251; ¦ n26+n57+n88+n119+n25=C ...pt261(=pt251); n24+n28+n57+n86+n120=C ...pt252; ¦ n30+n59+n88+n117+n21=C ...pt262(=pt222);

n20+n36+n57+n78+n124=C ...pt253; ¦ n46+n67+n88+n109+n5=C ...pt263(=pt103);

n19+n38+n57+n76+n125=C ...pt254; ¦ n50+n69+n88+n107+n1=C ...pt264(=pt74);

Fig.26 illustrates another style of Pan-magic Shift. It is transformed from the basic Fig.1, moving the vertical plane at the left side far beyond the right end. The next Fig.51 is transformed from the Fig.26 in the same way.

51----------76---------101---------- 1----------26 52----------77---------102---------- 2----------27 ¦52 77 102 2 ¦27 ¦53 78 103 3 ¦28

56 53 81 78 106 103 6 3 31 28 57 54 82 79 107 104 7 4 32 29

¦57 54 82 79 107 104 7 4 ¦32 29 ¦58 55 83 80 108 105 8 5 ¦33 30 61 58 55--86--83--80-111-108-105--11-- 8-- 5--36--33--30 62 59 51--87--84--76-112-109-101--12-- 9-- 1--37--34--26

¦62 59 ¦ 87 84 112 109 12 9 ¦37 34 ¦ ¦63 60 ¦ 88 85 113 110 13 10 ¦38 35 ¦

66 63 60 91 88 85 116 113 110 16 13 10 41 38 35 67 64 56 92 89 81 117 114 106 17 14 6 42 39 31 ¦67 64 ¦ 92 89 117 114 17 14 ¦42 39 ¦ ¦68 65 ¦ 93 90 118 115 18 15 ¦43 40 ¦

71--68--65--96--93--90-121-118-115--21--18--15--46 43 40 72--69--61--97--94--86-122-119-111--22--19--11--47 44 36

72 69 ¦ 97 94 122 119 22 19 47 44 ¦ 73 70 ¦ 98 95 123 120 23 20 48 45 ¦ 73 70 98 95 123 120 23 20 48 45 74 66 99 91 124 116 24 16 49 41

74 ¦ 99 124 24 49 ¦ 75 ¦ 100 125 25 50 ¦

75---------100---------125----------25----------50 71----------96---------121----------21----------46

5

n51+n82+n113+n19+n50=C ...pt511(=pt191); ¦ n52+n83+n114+n20+n46=C ...pt521(=pt201);


n71+n92+n113+n9+n30=C ...pt513(=pt143); ¦ n72+n93+n114+n10+n26=C ...pt523(=pt153); n75+n94+n113+n7+n26=C ...pt514(=pt134); ¦ n71+n95+n114+n8+n27=C ...pt524(=pt144);

There are three styles of Pan-magic Shift in all for our magic cube of any order: (1) From the back side to the front end, (2) From the top side down to the bottom end and (3) From the left side to the right end. But you may suppose opposite directions to every dimensional shifting: (4) From the front side to the back, (5) From the bottom side up to the top, (6) From the right side to the left. And you may combine any of those six styles one after another to make any different cubes from each other. 93---------118----------18----------43----------68 94---------119----------19----------44----------69

¦94 119 19 44 ¦69 ¦95 120 20 45 ¦70 98 95 123 120 23 20 48 45 73 70 99 91 124 116 24 16 49 41 74 66

¦99 91 124 116 24 16 49 41 ¦74 66 100 92 125 117 25 17 50 42 ¦75 67

78 100 92-103-125-117-- 3--25--17--28--50--42--53--75--67 79 96 93-104-121-118-- 4--21--18--29--46--43--54--71--68 ¦79 96 ¦ 104 121 4 21 29 46 ¦54 71 ¦ ¦80 97 ¦ 105 122 5 22 30 47 ¦55 72 ¦

83 80 97 108 105 122 8 5 22 33 30 47 58 55 72 84 76 98 109 101 123 9 1 23 34 26 48 59 51 73

¦84 76 ¦ 109 101 9 1 34 26 ¦59 51 ¦ ¦85 77 ¦ 110 102 10 2 35 27 ¦60 52 ¦ 88--85--77-113-110-102--13--10-- 2--38--35--27--63 60 52 89--81--78-114-106-103--14-- 6-- 3--39--31--28--64 56 53

89 81 ¦ 114 106 14 6 39 31 64 56 ¦ 90 82 ¦ 115 107 15 7 40 32 65 57 ¦

90 82 115 107 15 7 40 32 65 57 86 83 111 108 11 8 36 33 61 58 86 ¦ 111 11 36 61 ¦ 87 ¦ 112 12 37 62 ¦

87---------112----------12----------37----------62 88---------113----------13----------38----------63





101---------- 1----------26----------51----------76 102---------- 2----------27----------52----------77

102 2 27 52 ¦77 103 3 28 53 ¦78 106 103 6 3 31 28 56 53 81 78 107 104 7 4 32 29 57 54 82 79

107 104 7 4 32 29 57 54 ¦82 79 108 105 8 5 33 30 58 55 ¦83 80

111 108 105--11-- 8-- 5--36--33--30--61--58--55--86--83--80 112 109 101--12-- 9-- 1--37--34--26--62--59--51--87--84--76 112 109 ¦ 12 9 37 34 62 59 ¦87 84 ¦ 113 110 ¦ 13 10 38 35 63 60 ¦88 85 ¦

116 113 110 16 13 10 41 38 35 66 63 60 91 88 85 117 114 106 17 14 6 42 39 31 67 64 56 92 89 81

117 114 ¦ 17 14 42 39 67 64 ¦92 89 ¦ 118 115 ¦ 18 15 43 40 68 65 ¦93 90 ¦ 121-118-115--21--18--15--46--43--40--71--68--65--96 93 90 122-119-111--22--19--11--47--44--36--72--69--61--97 94 86

122 119 ¦ 22 19 47 44 72 69 97 94 ¦ 123 120 ¦ 23 20 48 45 73 70 98 95 ¦

123 120 23 20 48 45 73 70 98 95 124 116 24 16 49 41 74 66 99 91 124 ¦ 24 49 74 99 ¦ 125 ¦ 25 50 75 100 ¦

125----------25----------50----------75---------100 121----------21----------46----------71----------96


n105+n9+n38+n67+n96=C ...pt1012(=pt102); ¦ n101+n10+n39+n68+n97=C ...pt1022(=pt62); n121+n17+n38+n59+n80=C ...pt1013(=pt223); ¦ n122+n18+n39+n60+n76=C ...pt1023(=pt233);


. . . . .

How many different cubes in all can we make by these styles of Pan-magic Shift? Yes, 125 in all. How many different conditions in all do we have to define for the Pan-triagonals at the first definition stage, then? Do we need as many as 500(=125 x 4)? No, not so many. If we remove repetitions of the same contents as many as we can, we need only the 100(=25 x 4) different conditions for Pan-triagonals. With those 100 Pan-triagonal definitions, we can use any style of Pan-magic Shift anytime and make all the different 125 Pan-magic Cubes always just as we have wanted. This is the basic concept of our Pan-triagonal Magic Cubes of Order 5. 2. How to define this type of object with the Basic Form and Basic Equations: Let me show you the definitions with the diagram and list of equations as usual.

** Basic Conditions for Magic Cubes of Order 5: ** n1+n2+n3+n4+n5=C; ¦ n1+n6+n11+n16+n21=C; ¦ n1+n26+n51+n76+n101=C;

6

n6+n7+n8+n9+n10=C; ¦ n2+n7+n12+n17+n22=C; ¦ n2+n27+n52+n77+n102=C; n11+n12+n13+n14+n15=C; ¦ n3+n8+n13+n18+n23=C; ¦ n3+n28+n53+n78+n103=C; n16+n17+n18+n19+n20=C; ¦ n4+n9+n14+n19+n24=C; ¦ n4+n29+n54+n79+n104=C; n21+n22+n23+n24+n25=C; ¦ n5+n10+n15+n20+n25=C; ¦ n5+n30+n55+n80+n105=C; n26+n27+n28+n29+n30=C; ¦ n26+n31+n36+n41+n46=C; ¦ n6+n31+n56+n81+n106=C; n31+n32+n33+n34+n35=C; ¦ n27+n32+n37+n42+n47=C; ¦ n7+n32+n57+n82+n107=C; n36+n37+n38+n39+n40=C; ¦ n28+n33+n38+n43+n48=C; ¦ n8+n33+n58+n83+n108=C; n41+n42+n43+n44+n45=C; ¦ n29+n34+n39+n44+n49=C; ¦ n9+n34+n59+n84+n109=C; n46+n47+n48+n49+n50=C; ¦ n30+n35+n40+n45+n50=C; ¦ n10+n35+n60+n85+n110=C; n51+n52+n53+n54+n55=C; ¦ n51+n56+n61+n66+n71=C; ¦ n11+n36+n61+n86+n111=C; n56+n57+n58+n59+n60=C; ¦ n52+n57+n62+n67+n72=C; ¦ n12+n37+n62+n87+n112=C; n61+n62+n63+n64+n65=C; ¦ n53+n58+n63+n68+n73=C; ¦ n13+n38+n63+n88+n113=C; n66+n67+n68+n69+n70=C; ¦ n54+n59+n64+n69+n74=C; ¦ n14+n39+n64+n89+n114=C; n71+n72+n73+n74+n75=C; ¦ n55+n60+n65+n70+n75=C; ¦ n15+n40+n65+n90+n115=C; n76+n77+n78+n79+n80=C; ¦ n76+n81+n86+n91+n96=C; ¦ n16+n41+n66+n91+n116=C; n81+n82+n83+n84+n85=C; ¦ n77+n82+n87+n92+n97=C; ¦ n17+n42+n67+n92+n117=C; n86+n87+n88+n89+n90=C; ¦ n78+n83+n88+n93+n98=C; ¦ n18+n43+n68+n93+n118=C; n91+n92+n93+n94+n95=C; ¦ n79+n84+n89+n94+n99=C; ¦ n19+n44+n69+n94+n119=C; n96+n97+n98+n99+n100=C; ¦ n80+n85+n90+n95+n100=C; ¦ n20+n45+n70+n95+n120=C; n101+n102+n103+n104+n105=C; ¦ n101+n106+n111+n116+n121=C; ¦ n21+n46+n71+n96+n121=C; n106+n107+n108+n109+n110=C; ¦ n102+n107+n112+n117+n122=C; ¦ n22+n47+n72+n97+n122=C; n111+n112+n113+n114+n115=C; ¦ n103+n108+n113+n118+n123=C; ¦ n23+n48+n73+n98+n123=C; n116+n117+n118+n119+n120=C; ¦ n104+n109+n114+n119+n124=C; ¦ n24+n49+n74+n99+n124=C; n121+n122+n123+n124+n125=C; ¦ n105+n110+n115+n120+n125=C; ¦ n25+n50+n75+n100+n125=C;

** Basic Form for Magic Cubes of Order 5 ** 1-------------- 26-------------- 51-------------- 76--------------101 ¦ ¦ ¦ 2 27 52 77 ¦102 6 31 56 81 106 ¦ 3 28 53 78 ¦ 103 ¦ 7 32 57 82 ¦107 11 4 36 29 61 54 86 79 111 104 ¦ 8 33 58 83 ¦ 108 ¦ 12 5----- 37------ 30----- 62------ 55----- 87------ 80-----112------105 16 9 ¦ 41 34 66 59 91 84 116 109 ¦ ¦ 13 ¦ 38 63 88 ¦ 113 ¦ ¦ 17 10 42 35 67 60 92 85 ¦117 110 21------ 14--¦-- 46------ 39----- 71------ 64----- 96------ 89-----121 114 ¦ 18 ¦ 43 68 93 118 ¦ 22 15 47 40 72 65 97 90 122 115 19 ¦ 44 69 94 119 ¦ 23 ¦ 48 73 98 123 ¦ 20 45 70 95 120 24 ¦ 49 74 99 124 ¦ ¦ ¦ 25-------------- 50-------------- 75--------------100--------------125

** The 100 Pan-triagonal Conditions for Pan-magic Cubes of Order 5: ** n1+n32+n63+n94+n125=C ...pt11; ¦ n5+n34+n63+n92+n121=C ...pt12; n21+n42+n63+n84+n105=C ...pt13; ¦ n25+n44+n63+n82+n101=C ...pt14; n2+n33+n64+n95+n121=C ...pt21; ¦ n1+n35+n64+n93+n122=C ...pt22; n22+n43+n64+n85+n101=C ...pt23; ¦ n21+n45+n64+n83+n102=C ...pt24; n3+n34+n65+n91+n122=C ...pt31; ¦ n2+n31+n65+n94+n123=C ...pt32; n23+n44+n65+n81+n102=C ...pt33; ¦ n22+n41+n65+n84+n103=C ...pt34; n4+n35+n61+n92+n123=C ...pt41; ¦ n3+n32+n61+n95+n124=C ...pt42; n24+n45+n61+n82+n103=C ...pt43; ¦ n23+n42+n61+n85+n104=C ...pt44; n5+n31+n62+n93+n124=C ...pt51; ¦ n4+n33+n62+n91+n125=C ...pt52; n25+n41+n62+n83+n104=C ...pt53; ¦ n24+n43+n62+n81+n105=C ...pt54; n6+n37+n68+n99+n105=C ...pt61; ¦ n10+n39+n68+n97+n101=C ...pt62; n1+n47+n68+n89+n110=C ...pt63; ¦ n5+n49+n68+n87+n106=C ...pt64; n7+n38+n69+n100+n101=C ...pt71; ¦ n6+n40+n69+n98+n102=C ...pt72; n2+n48+n69+n90+n106=C ...pt73; ¦ n1+n50+n69+n88+n107=C ...pt74;

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n8+n39+n70+n96+n102=C ...pt81; ¦ n7+n36+n70+n99+n103=C ...pt82; n3+n49+n70+n86+n107=C ...pt83; ¦ n2+n46+n70+n89+n108=C ...pt84; n9+n40+n66+n97+n103=C ...pt91; ¦ n8+n37+n66+n100+n104=C ...pt92; n4+n50+n66+n87+n108=C ...pt93; ¦ n3+n47+n66+n90+n109=C ...pt94; n10+n36+n67+n98+n104=C ...pt101; ¦ n9+n38+n67+n96+n105=C ...pt102; n5+n46+n67+n88+n109=C ...pt103; ¦ n4+n48+n67+n86+n110=C ...pt104; n11+n42+n73+n79+n110=C ...pt111; ¦ n15+n44+n73+n77+n106=C ...pt112; n6+n27+n73+n94+n115=C ...pt113; ¦ n10+n29+n73+n92+n111=C ...pt114; n12+n43+n74+n80+n106=C ...pt121; ¦ n11+n45+n74+n78+n107=C ...pt122; n7+n28+n74+n95+n111=C ...pt123; ¦ n6+n30+n74+n93+n112=C ...pt124; n13+n44+n75+n76+n107=C ...pt131; ¦ n12+n41+n75+n79+n108=C ...pt132; n8+n29+n75+n91+n112=C ...pt133; ¦ n7+n26+n75+n94+n113=C ...pt134; n14+n45+n71+n77+n108=C ...pt141; ¦ n13+n42+n71+n80+n109=C ...pt142; n9+n30+n71+n92+n113=C ...pt143; ¦ n8+n27+n71+n95+n114=C ...pt144; n15+n41+n72+n78+n109=C ...pt151; ¦ n14+n43+n72+n76+n110=C ...pt152; n10+n26+n72+n93+n114=C ...pt153; ¦ n9+n28+n72+n91+n115=C ...pt154; n16+n47+n53+n84+n115=C ...pt161; ¦ n20+n49+n53+n82+n111=C ...pt162; n11+n32+n53+n99+n120=C ...pt163; ¦ n15+n34+n53+n97+n116=C ...pt164; n17+n48+n54+n85+n111=C ...pt171; ¦ n16+n50+n54+n83+n112=C ...pt172; n12+n33+n54+n100+n116=C ...pt173; ¦ n11+n35+n54+n98+n117=C ...pt174; n18+n49+n55+n81+n112=C ...pt181; ¦ n17+n46+n55+n84+n113=C ...pt182; n13+n34+n55+n96+n117=C ...pt183; ¦ n12+n31+n55+n99+n118=C ...pt184; n19+n50+n51+n82+n113=C ...pt191; ¦ n18+n47+n51+n85+n114=C ...pt192; n14+n35+n51+n97+n118=C ...pt193; ¦ n13+n32+n51+n100+n119=C ...pt194; n20+n46+n52+n83+n114=C ...pt201; ¦ n19+n48+n52+n81+n115=C ...pt202; n15+n31+n52+n98+n119=C ...pt203; ¦ n14+n33+n52+n96+n120=C ...pt204; n21+n27+n58+n89+n120=C ...pt211; ¦ n25+n29+n58+n87+n116=C ...pt212; n16+n37+n58+n79+n125=C ...pt213; ¦ n20+n39+n58+n77+n121=C ...pt214; n22+n28+n59+n90+n116=C ...pt221; ¦ n21+n30+n59+n88+n117=C ...pt222; n17+n38+n59+n80+n121=C ...pt223; ¦ n16+n40+n59+n78+n122=C ...pt224; n23+n29+n60+n86+n117=C ...pt231; ¦ n22+n26+n60+n89+n118=C ...pt232; n18+n39+n60+n76+n122=C ...pt233; ¦ n17+n36+n60+n79+n123=C ...pt234; n24+n30+n56+n87+n118=C ...pt241; ¦ n23+n27+n56+n90+n119=C ...pt242; n19+n40+n56+n77+n123=C ...pt243; ¦ n18+n37+n56+n80+n124=C ...pt244; n25+n26+n57+n88+n119=C ...pt251; ¦ n24+n28+n57+n86+n120=C ...pt252; n20+n36+n57+n78+n124=C ...pt253; ¦ n19+n38+n57+n76+n125=C ...pt254; This time we don't have anything like Self-complementary Pairs to be set at the same time in every single procedure. We have nothing like the invariable value for the variable n63 at the geometrical center of our cube, either. n63 may take any value now. 3. How to make our 'Latin Cubes' of Order 5 for the "Complete Euler Type": Let's build the "Complete Euler Cubes" of Order 5 for the Pan-triagonal type here, using the Positional Number System of Base 5 (5-th Increment Number System). Before all we have to make 'Latin Cubes' of Order 5, to combine any three of them and suppose the set to be the Simulated Decomposed Layer Units of our object. The Latin Cubes are defined almost in the same way as above, but you must only change the values of Constant Sums as C=10 and SC=4. On top of that, Latin Cubes must always be made under such the strict rules as: (1) Any Latin Cube must be simply made up of 5 numerical letters: {0, 1, 2, 3 and 4}. Each letter should appear as often as 25 times, no more and no less.

(2) Each row, each column and each pillar of Latin Cubes must be made up of {0, 1, 2, 3 and 4} using each number strictly once. Neither duplication nor lack of any number must be found in it. These strict conditions should make the constant sum of each row, column and pillar of our final object always realized: ((0+1+2+3+4)x5(Dec)+(0+1+2+3+4))x5(Dec)+(0+1+2+3+4)=310(Dec)

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(in Mathematical Expression of Cube)

(3) Each of the Pan-triagonals must also be made up of 5 numbers: {0, 1, 2, 3 and 4} using each one strictly once. Neither duplication nor lack of any number is permitted. 4. What the 'Latin Cubes' look like: The next list shows some sample solutions of Latin Cubes of Order 5 for the Pan-triagonal Type, made under the Positional Number System of Base 5. 1/ 25/ 49/ 0-----1-----2-----3-----4 0-----2-----1-----3-----4 0-----3-----1-----2-----4 ¦1 2 3 4 ¦0 ¦2 1 3 4 ¦0 ¦3 1 2 4 ¦0 1 2 2 3 3 4 4 0 0 1 2 1 1 3 3 4 4 0 0 2 3 1 1 2 2 4 4 0 0 3 ¦2 3 3 4 4 0 0 1 ¦1 2 ¦1 3 3 4 4 0 0 2 ¦2 1 ¦1 2 2 4 4 0 0 3 ¦3 1 2 3 4-3-4-0-4-0-1-0-1-2-1-2-3 1 3 4-3-4-0-4-0-2-0-2-1-2-1-3 1 2 4-2-4-0-4-0-3-0-3-1-3-1-2 ¦3 4¦ 4 0 0 1 1 2 ¦2 3¦ ¦3 4¦ 4 0 0 2 2 1 ¦1 3¦ ¦2 4¦ 4 0 0 3 3 1 ¦1 2¦ 3 4 0 4 0 1 0 1 2 1 2 3 2 3 4 3 4 0 4 0 2 0 2 1 2 1 3 1 3 4 2 4 0 4 0 3 0 3 1 3 1 2 1 2 4 ¦4 0¦ 0 1 1 2 2 3 ¦3 4¦ ¦4 0¦ 0 2 2 1 1 3 ¦3 4¦ ¦4 0¦ 0 3 3 1 1 2 ¦2 4¦ 4-0-1-0-1-2-1-2-3-2-3-4-3 4 0 4-0-2-0-2-1-2-1-3-1-3-4-3 4 0 4-0-3-0-3-1-3-1-2-1-2-4-2 4 0 0 1¦ 1 2 2 3 3 4 4 0¦ 0 2¦ 2 1 1 3 3 4 4 0¦ 0 3¦ 3 1 1 2 2 4 4 0¦ 1 2 2 3 3 4 4 0 0 1 2 1 1 3 3 4 4 0 0 2 3 1 1 2 2 4 4 0 0 3 2¦ 3 4 0 1¦ 1¦ 3 4 0 2¦ 1¦ 2 4 0 3¦ 3-----4-----0-----1-----2 3-----4-----0-----2-----1 2-----4-----0-----3-----1 I want you to watch these samples and check if each of them is successfully made under those strict conditions above. Will you especially examine each content of rows, columns, pillars and all pan-triagonals? And finally check every two adjacent numbers sitting next in all seven lines to see whether they are the same or not. 5. How to make the Latin Cubes of Order 5 How can we make such the Latin Cubes as shown above, then? How many of them can we make in all? I plan to make them here by our "New Euler's Method" under the Positional Number System of Base 5. Since we can only set only one variable in each step, we must have 125 steps in all for all variables consequently. We must also check whether each variable obeys the strict Latin Rule or not, precisely on all the seven lines such as each row, column, pillar and every four pan-triagonals in each step. It should be very hard for us to write such a long, complicated program, as you guess. It is the fact that I had to develop my own new technique, so called "Computer Assisted Programming". It makes my computer assist me to dictate the program routines with rich contents as accurate as possible. I wrote a kind of 'meta-program' that should dictate the actual program we need for our purpose. Let me list out only the core part of my actual program here as follows. /**/ short cnt0, cnt1, cnt2, cnt3; short nm[126]; short bc[176][5]; //For the 25 Rows, 25 Columns, 25 Pillars and 100 Pan-triagonals short d[4][129]; //For the Contemporary Data for listing out 3 Answers short tluh[481]; //For the selection of Standard Solutions checking the High Layer short tlu[481][126]; //For storing all the Latin Units Composed char mtc[481][481]; //For the Compatibility Data of any two Units /**/ /** Program Modules for 'Latin Cubes' 5x5x5 of the Pan-triagonal Type dictated by Kanji Setsuda on May 30, 2010 with MacOSX & Xcode3.1.2 **/ /* Level 1: */ /* Set N1 */ void stp01(){ short a;

9

for(a=0;a<5;a++){ if((bc[1][a]==0)&&(bc[2][a]==0)&&(bc[3][a]==0)&&(bc[76][a]==0)){ if((bc[81][a]==0)&&(bc[98][a]==0)&&(bc[103][a]==0)){ nm[1]=a; bc[1][a]=1; bc[2][a]=1; bc[3][a]=1; bc[76][a]=1; bc[81][a]=1; bc[98][a]=1; bc[103][a]=1; stp02(); bc[1][a]=0; bc[2][a]=0; bc[3][a]=0; bc[76][a]=0; bc[81][a]=0; bc[98][a]=0; bc[103][a]=0;}} } } /* Set N2 */ void stp02(){ short a; for(a=0;a<5;a++){ if((bc[1][a]==0)&&(bc[5][a]==0)&&(bc[6][a]==0)&&(bc[80][a]==0)){ if((bc[85][a]==0)&&(bc[102][a]==0)&&(bc[107][a]==0)){ nm[2]=a; bc[1][a]=1; bc[5][a]=1; bc[6][a]=1; bc[80][a]=1; bc[85][a]=1; bc[102][a]=1; bc[107][a]=1; cnt2=0; stp03(); bc[1][a]=0; bc[5][a]=0; bc[6][a]=0; bc[80][a]=0; bc[85][a]=0; bc[102][a]=0; bc[107][a]=0;}} } } /* Set N3 */ void stp03(){ short a; for(a=0;a<5;a++){ if((bc[1][a]==0)&&(bc[8][a]==0)&&(bc[9][a]==0)&&(bc[84][a]==0)){ if((bc[89][a]==0)&&(bc[106][a]==0)&&(bc[111][a]==0)){ nm[3]=a; bc[1][a]=1; bc[8][a]=1; bc[9][a]=1; bc[84][a]=1; bc[89][a]=1; bc[106][a]=1; bc[111][a]=1; stp04(); bc[1][a]=0; bc[8][a]=0; bc[9][a]=0; bc[84][a]=0; bc[89][a]=0; bc[106][a]=0; bc[111][a]=0;}} } } /* Set N4 */ void stp04(){ short a; for(a=0;a<5;a++){ if((bc[1][a]==0)&&(bc[11][a]==0)&&(bc[12][a]==0)&&(bc[88][a]==0)){ if((bc[93][a]==0)&&(bc[110][a]==0)&&(bc[115][a]==0)){ nm[4]=a; bc[1][a]=1; bc[11][a]=1; bc[12][a]=1; bc[88][a]=1; bc[93][a]=1; bc[110][a]=1; bc[115][a]=1; stp05(); bc[1][a]=0; bc[11][a]=0; bc[12][a]=0; bc[88][a]=0; bc[93][a]=0; bc[110][a]=0; bc[115][a]=0;}} } } /* Set N5 */ void stp05(){ short a; for(a=0;a<5;a++){ if((bc[1][a]==0)&&(bc[14][a]==0)&&(bc[15][a]==0)&&(bc[77][a]==0)){ if((bc[92][a]==0)&&(bc[99][a]==0)&&(bc[114][a]==0)){ nm[5]=a; bc[1][a]=1; bc[14][a]=1; bc[15][a]=1; bc[77][a]=1; bc[92][a]=1; bc[99][a]=1; bc[114][a]=1; stp06(); bc[1][a]=0; bc[14][a]=0; bc[15][a]=0; bc[77][a]=0; bc[92][a]=0; bc[99][a]=0; bc[114][a]=0;}}

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} } /* Set N6 */ void stp06(){ short a; for(a=0;a<5;a++){ if((bc[2][a]==0)&&(bc[4][a]==0)&&(bc[18][a]==0)&&(bc[96][a]==0)){ if((bc[101][a]==0)&&(bc[118][a]==0)&&(bc[123][a]==0)){ nm[6]=a; bc[2][a]=1; bc[4][a]=1; bc[18][a]=1; bc[96][a]=1; bc[101][a]=1; bc[118][a]=1; bc[123][a]=1; stp07(); bc[2][a]=0; bc[4][a]=0; bc[18][a]=0; bc[96][a]=0; bc[101][a]=0; bc[118][a]=0; bc[123][a]=0;}} } } /* Set N7 */ void stp07(){ short a; for(a=0;a<5;a++){ if((bc[4][a]==0)&&(bc[5][a]==0)&&(bc[21][a]==0)&&(bc[100][a]==0)){ if((bc[105][a]==0)&&(bc[122][a]==0)&&(bc[127][a]==0)){ nm[7]=a; bc[4][a]=1; bc[5][a]=1; bc[21][a]=1; bc[100][a]=1; bc[105][a]=1; bc[122][a]=1; bc[127][a]=1; stp08(); bc[4][a]=0; bc[5][a]=0; bc[21][a]=0; bc[100][a]=0; bc[105][a]=0; bc[122][a]=0; bc[127][a]=0;}} } } /* Set N8 */ void stp08(){ short a; for(a=0;a<5;a++){ if((bc[4][a]==0)&&(bc[8][a]==0)&&(bc[24][a]==0)&&(bc[104][a]==0)){ if((bc[109][a]==0)&&(bc[126][a]==0)&&(bc[131][a]==0)){ nm[8]=a; bc[4][a]=1; bc[8][a]=1; bc[24][a]=1; bc[104][a]=1; bc[109][a]=1; bc[126][a]=1; bc[131][a]=1; stp09(); bc[4][a]=0; bc[8][a]=0; bc[24][a]=0; bc[104][a]=0; bc[109][a]=0; bc[126][a]=0; bc[131][a]=0;}} } } /* Set N9 */ void stp09(){ short a; for(a=0;a<5;a++){ if((bc[4][a]==0)&&(bc[11][a]==0)&&(bc[27][a]==0)&&(bc[108][a]==0)){ if((bc[113][a]==0)&&(bc[130][a]==0)&&(bc[135][a]==0)){ nm[9]=a; bc[4][a]=1; bc[11][a]=1; bc[27][a]=1; bc[108][a]=1; bc[113][a]=1; bc[130][a]=1; bc[135][a]=1; stp10(); bc[4][a]=0; bc[11][a]=0; bc[27][a]=0; bc[108][a]=0; bc[113][a]=0; bc[130][a]=0; bc[135][a]=0;}} } } /* Set N10 */ void stp10(){ short a; for(a=0;a<5;a++){ if((bc[4][a]==0)&&(bc[14][a]==0)&&(bc[30][a]==0)&&(bc[97][a]==0)){ if((bc[112][a]==0)&&(bc[119][a]==0)&&(bc[134][a]==0)){ nm[10]=a;

11

bc[4][a]=1; bc[14][a]=1; bc[30][a]=1; bc[97][a]=1; bc[112][a]=1; bc[119][a]=1; bc[134][a]=1; stp11(); bc[4][a]=0; bc[14][a]=0; bc[30][a]=0; bc[97][a]=0; bc[112][a]=0; bc[119][a]=0; bc[134][a]=0;}} } } /* Set N11 */ void stp11(){ short a; for(a=0;a<5;a++){ if((bc[2][a]==0)&&(bc[7][a]==0)&&(bc[33][a]==0)&&(bc[116][a]==0)){ if((bc[121][a]==0)&&(bc[138][a]==0)&&(bc[143][a]==0)){ nm[11]=a; bc[2][a]=1; bc[7][a]=1; bc[33][a]=1; bc[116][a]=1; bc[121][a]=1; bc[138][a]=1; bc[143][a]=1; stp12(); bc[2][a]=0; bc[7][a]=0; bc[33][a]=0; bc[116][a]=0; bc[121][a]=0; bc[138][a]=0; bc[143][a]=0;}} } } /* Set N12 */ void stp12(){ short a; for(a=0;a<5;a++){ if((bc[5][a]==0)&&(bc[7][a]==0)&&(bc[36][a]==0)&&(bc[120][a]==0)){ if((bc[125][a]==0)&&(bc[142][a]==0)&&(bc[147][a]==0)){ nm[12]=a; bc[5][a]=1; bc[7][a]=1; bc[36][a]=1; bc[120][a]=1; bc[125][a]=1; bc[142][a]=1; bc[147][a]=1; stp13(); bc[5][a]=0; bc[7][a]=0; bc[36][a]=0; bc[120][a]=0; bc[125][a]=0; bc[142][a]=0; bc[147][a]=0;}} } } /* Set N13 */ void stp13(){ short a; for(a=0;a<5;a++){ if((bc[7][a]==0)&&(bc[8][a]==0)&&(bc[39][a]==0)&&(bc[124][a]==0)){ if((bc[129][a]==0)&&(bc[146][a]==0)&&(bc[151][a]==0)){ nm[13]=a; bc[7][a]=1; bc[8][a]=1; bc[39][a]=1; bc[124][a]=1; bc[129][a]=1; bc[146][a]=1; bc[151][a]=1; stp14(); bc[7][a]=0; bc[8][a]=0; bc[39][a]=0; bc[124][a]=0; bc[129][a]=0; bc[146][a]=0; bc[151][a]=0;}} } } /* Set N14 */ void stp14(){ short a; for(a=0;a<5;a++){ if((bc[7][a]==0)&&(bc[11][a]==0)&&(bc[42][a]==0)&&(bc[128][a]==0)){ if((bc[133][a]==0)&&(bc[150][a]==0)&&(bc[155][a]==0)){ nm[14]=a; bc[7][a]=1; bc[11][a]=1; bc[42][a]=1; bc[128][a]=1; bc[133][a]=1; bc[150][a]=1; bc[155][a]=1; stp15(); bc[7][a]=0; bc[11][a]=0; bc[42][a]=0; bc[128][a]=0; bc[133][a]=0; bc[150][a]=0; bc[155][a]=0;}} } } /* Set N15 */ void stp15(){

12

short a; for(a=0;a<5;a++){ if((bc[7][a]==0)&&(bc[14][a]==0)&&(bc[45][a]==0)&&(bc[117][a]==0)){ if((bc[132][a]==0)&&(bc[139][a]==0)&&(bc[154][a]==0)){ nm[15]=a; bc[7][a]=1; bc[14][a]=1; bc[45][a]=1; bc[117][a]=1; bc[132][a]=1; bc[139][a]=1; bc[154][a]=1; stp16(); bc[7][a]=0; bc[14][a]=0; bc[45][a]=0; bc[117][a]=0; bc[132][a]=0; bc[139][a]=0; bc[154][a]=0;}} } } /* Set N16 */ void stp16(){ short a; for(a=0;a<5;a++){ if((bc[2][a]==0)&&(bc[10][a]==0)&&(bc[48][a]==0)&&(bc[136][a]==0)){ if((bc[141][a]==0)&&(bc[158][a]==0)&&(bc[163][a]==0)){ nm[16]=a; bc[2][a]=1; bc[10][a]=1; bc[48][a]=1; bc[136][a]=1; bc[141][a]=1; bc[158][a]=1; bc[163][a]=1; stp17(); bc[2][a]=0; bc[10][a]=0; bc[48][a]=0; bc[136][a]=0; bc[141][a]=0; bc[158][a]=0; bc[163][a]=0;}} } } /* Set N17 */ void stp17(){ short a; for(a=0;a<5;a++){ if((bc[5][a]==0)&&(bc[10][a]==0)&&(bc[51][a]==0)&&(bc[140][a]==0)){ if((bc[145][a]==0)&&(bc[162][a]==0)&&(bc[167][a]==0)){ nm[17]=a; bc[5][a]=1; bc[10][a]=1; bc[51][a]=1; bc[140][a]=1; bc[145][a]=1; bc[162][a]=1; bc[167][a]=1; stp18(); bc[5][a]=0; bc[10][a]=0; bc[51][a]=0; bc[140][a]=0; bc[145][a]=0; bc[162][a]=0; bc[167][a]=0;}} } } /* Set N18 */ void stp18(){ short a; for(a=0;a<5;a++){ if((bc[8][a]==0)&&(bc[10][a]==0)&&(bc[54][a]==0)&&(bc[144][a]==0)){ if((bc[149][a]==0)&&(bc[166][a]==0)&&(bc[171][a]==0)){ nm[18]=a; bc[8][a]=1; bc[10][a]=1; bc[54][a]=1; bc[144][a]=1; bc[149][a]=1; bc[166][a]=1; bc[171][a]=1; stp19(); bc[8][a]=0; bc[10][a]=0; bc[54][a]=0; bc[144][a]=0; bc[149][a]=0; bc[166][a]=0; bc[171][a]=0;}} } } /* Set N19 */ void stp19(){ short a; for(a=0;a<5;a++){ if((bc[10][a]==0)&&(bc[11][a]==0)&&(bc[57][a]==0)&&(bc[148][a]==0)){ if((bc[153][a]==0)&&(bc[170][a]==0)&&(bc[175][a]==0)){ nm[19]=a; bc[10][a]=1; bc[11][a]=1; bc[57][a]=1; bc[148][a]=1; bc[153][a]=1; bc[170][a]=1; bc[175][a]=1; stp20(); bc[10][a]=0; bc[11][a]=0; bc[57][a]=0; bc[148][a]=0;

13

bc[153][a]=0; bc[170][a]=0; bc[175][a]=0;}} } } /* Set N20 */ void stp20(){ short a; for(a=0;a<5;a++){ if((bc[10][a]==0)&&(bc[14][a]==0)&&(bc[60][a]==0)&&(bc[137][a]==0)){ if((bc[152][a]==0)&&(bc[159][a]==0)&&(bc[174][a]==0)){ nm[20]=a; bc[10][a]=1; bc[14][a]=1; bc[60][a]=1; bc[137][a]=1; bc[152][a]=1; bc[159][a]=1; bc[174][a]=1; stp21(); bc[10][a]=0; bc[14][a]=0; bc[60][a]=0; bc[137][a]=0; bc[152][a]=0; bc[159][a]=0; bc[174][a]=0;}} } } /* Set N21 */ void stp21(){ short a; for(a=0;a<5;a++){ if((bc[2][a]==0)&&(bc[13][a]==0)&&(bc[63][a]==0)&&(bc[78][a]==0)){ if((bc[83][a]==0)&&(bc[156][a]==0)&&(bc[161][a]==0)){ nm[21]=a; bc[2][a]=1; bc[13][a]=1; bc[63][a]=1; bc[78][a]=1; bc[83][a]=1; bc[156][a]=1; bc[161][a]=1; stp22(); bc[2][a]=0; bc[13][a]=0; bc[63][a]=0; bc[78][a]=0; bc[83][a]=0; bc[156][a]=0; bc[161][a]=0;}} } } /* Set N22 */ void stp22(){ short a; for(a=0;a<5;a++){ if((bc[5][a]==0)&&(bc[13][a]==0)&&(bc[66][a]==0)&&(bc[82][a]==0)){ if((bc[87][a]==0)&&(bc[160][a]==0)&&(bc[165][a]==0)){ nm[22]=a; bc[5][a]=1; bc[13][a]=1; bc[66][a]=1; bc[82][a]=1; bc[87][a]=1; bc[160][a]=1; bc[165][a]=1; stp23(); bc[5][a]=0; bc[13][a]=0; bc[66][a]=0; bc[82][a]=0; bc[87][a]=0; bc[160][a]=0; bc[165][a]=0;}} } } /* Set N23 */ void stp23(){ short a; for(a=0;a<5;a++){ if((bc[8][a]==0)&&(bc[13][a]==0)&&(bc[69][a]==0)&&(bc[86][a]==0)){ if((bc[91][a]==0)&&(bc[164][a]==0)&&(bc[169][a]==0)){ nm[23]=a; bc[8][a]=1; bc[13][a]=1; bc[69][a]=1; bc[86][a]=1; bc[91][a]=1; bc[164][a]=1; bc[169][a]=1; stp24(); bc[8][a]=0; bc[13][a]=0; bc[69][a]=0; bc[86][a]=0; bc[91][a]=0; bc[164][a]=0; bc[169][a]=0;}} } } /* Set N24 */ void stp24(){ short a; for(a=0;a<5;a++){ if((bc[11][a]==0)&&(bc[13][a]==0)&&(bc[72][a]==0)&&(bc[90][a]==0)){ if((bc[95][a]==0)&&(bc[168][a]==0)&&(bc[173][a]==0)){

14

nm[24]=a; bc[11][a]=1; bc[13][a]=1; bc[72][a]=1; bc[90][a]=1; bc[95][a]=1; bc[168][a]=1; bc[173][a]=1; stp25(); bc[11][a]=0; bc[13][a]=0; bc[72][a]=0; bc[90][a]=0; bc[95][a]=0; bc[168][a]=0; bc[173][a]=0;}} } } /* Set N25 */ void stp25(){ short a; for(a=0;a<5;a++){ if((bc[13][a]==0)&&(bc[14][a]==0)&&(bc[75][a]==0)&&(bc[79][a]==0)){ if((bc[94][a]==0)&&(bc[157][a]==0)&&(bc[172][a]==0)){ nm[25]=a; bc[13][a]=1; bc[14][a]=1; bc[75][a]=1; bc[79][a]=1; bc[94][a]=1; bc[157][a]=1; bc[172][a]=1; stp26(); bc[13][a]=0; bc[14][a]=0; bc[75][a]=0; bc[79][a]=0; bc[94][a]=0; bc[157][a]=0; bc[172][a]=0;}} } } /* Level 2: */ /* Set N26 */ void stp26(){ short a; for(a=0;a<5;a++){ if((bc[3][a]==0)&&(bc[16][a]==0)&&(bc[17][a]==0)&&(bc[127][a]==0)){ if((bc[134][a]==0)&&(bc[165][a]==0)&&(bc[172][a]==0)){ nm[26]=a; bc[3][a]=1; bc[16][a]=1; bc[17][a]=1; bc[127][a]=1; bc[134][a]=1; bc[165][a]=1; bc[172][a]=1; stp27(); bc[3][a]=0; bc[16][a]=0; bc[17][a]=0; bc[127][a]=0; bc[134][a]=0; bc[165][a]=0; bc[172][a]=0;}} } } /**/ /* ...(skip)... */ /**/ /* Set N124 */ void stp124(){ short a; for(a=0;a<5;a++){ if((bc[71][a]==0)&&(bc[72][a]==0)&&(bc[73][a]==0)&&(bc[89][a]==0)){ if((bc[92][a]==0)&&(bc[171][a]==0)&&(bc[174][a]==0)){ nm[124]=a; bc[71][a]=1; bc[72][a]=1; bc[73][a]=1; bc[89][a]=1; bc[92][a]=1; bc[171][a]=1; bc[174][a]=1; stp125(); bc[71][a]=0; bc[72][a]=0; bc[73][a]=0; bc[89][a]=0; bc[92][a]=0; bc[171][a]=0; bc[174][a]=0;}} } } /* Set N125 */ void stp125(){ short a; for(a=0;a<5;a++){ if((bc[73][a]==0)&&(bc[74][a]==0)&&(bc[75][a]==0)&&(bc[76][a]==0)){ if((bc[93][a]==0)&&(bc[158][a]==0)&&(bc[175][a]==0)){ nm[125]=a; bc[73][a]=1; bc[74][a]=1; bc[75][a]=1; bc[76][a]=1; bc[93][a]=1; bc[158][a]=1; bc[175][a]=1; recordans(); bc[73][a]=0; bc[74][a]=0; bc[75][a]=0; bc[76][a]=0;

15

bc[93][a]=0; bc[158][a]=0; bc[175][a]=0;}} } } /**/ /* Record the Latin Units */ void recordans(){ short n; tlu[cnt0][0]=cnt0+1; for(n=1;n<126;n++){tlu[cnt0][n]=nm[n];} if((nm[1]<=nm[5])&&(nm[1]<=nm[21])&&(nm[1]<=nm[25])&&(nm[1]<=nm[101])&&(nm[1]<=nm[105])){ if((nm[1]<=nm[121])&&(nm[1]<=nm[125])&&(nm[2]<=nm[6])&&(nm[6]<=nm[26])){tluh[cnt0]++; cnt1++;}} cnt0++; cnt2++; if(cnt2==1){ d[cnt3][0]=cnt0; d[cnt3+1][0]=0; for(n=1;n<126;n++){d[cnt3][n]=nm[n];} cnt3++; if(cnt3==3){prdlu(cnt3); cnt3=0;} } } /**/ /* Make the Reference Table */ void mkreftbl(){ short m,l,n,t,d; /* Measure How Similar each Pair is. */ for(m=0;m<cnt0;m++){ for(l=0;l<cnt0;l++){ t=0; for(n=1;n<126;n++){if(tlu[m][n]==tlu[l][n]){t++;}} mtc[m][l]=t; t=0; for(n=1;n<126;n++){if(tlu[m][n]+tlu[l][n]==4){t++;}} if(mtc[m][l]<t){mtc[m][l]=t;} }} } /**/ In every step we always watch each variable on the seven lines: row, column, pillar and 4 pan-triagonals and check whether each value on it is 'Used' or 'Un-used' there. If it is 'Used', then we must have another value to be checked again. But, if it is all 'Un-used' in the seven lines, then we can take it for the variable now and go on to the next step, after we set the flags for the value as 'Used' in all the seven directions. 6. How many Latin Units we could have in all: The compact list below shows the recent result of my research, though only a part. The total count of Latin Units is 480. What a small count it is! But it means we have to check every combination of three Latin Units in over 110 million ways(=480 x 480 x 480)! Why so many! ** 'Latin Cubes' of Order 5 for Pan-triagonal Type: ** ** by Kanji Setsuda on June 7, 2010 with Xcode3.1.2 ** 1/ 25/ 49/ 0-----1-----2-----3-----4 0-----2-----1-----3-----4 0-----3-----1-----2-----4 ¦1 2 3 4 ¦0 ¦2 1 3 4 ¦0 ¦3 1 2 4 ¦0 1 2 2 3 3 4 4 0 0 1 2 1 1 3 3 4 4 0 0 2 3 1 1 2 2 4 4 0 0 3 ¦2 3 3 4 4 0 0 1 ¦1 2 ¦1 3 3 4 4 0 0 2 ¦2 1 ¦1 2 2 4 4 0 0 3 ¦3 1 2 3 4-3-4-0-4-0-1-0-1-2-1-2-3 1 3 4-3-4-0-4-0-2-0-2-1-2-1-3 1 2 4-2-4-0-4-0-3-0-3-1-3-1-2 ¦3 4¦ 4 0 0 1 1 2 ¦2 3¦ ¦3 4¦ 4 0 0 2 2 1 ¦1 3¦ ¦2 4¦ 4 0 0 3 3 1 ¦1 2¦ 3 4 0 4 0 1 0 1 2 1 2 3 2 3 4 3 4 0 4 0 2 0 2 1 2 1 3 1 3 4 2 4 0 4 0 3 0 3 1 3 1 2 1 2 4 ¦4 0¦ 0 1 1 2 2 3 ¦3 4¦ ¦4 0¦ 0 2 2 1 1 3 ¦3 4¦ ¦4 0¦ 0 3 3 1 1 2 ¦2 4¦ 4-0-1-0-1-2-1-2-3-2-3-4-3 4 0 4-0-2-0-2-1-2-1-3-1-3-4-3 4 0 4-0-3-0-3-1-3-1-2-1-2-4-2 4 0 0 1¦ 1 2 2 3 3 4 4 0¦ 0 2¦ 2 1 1 3 3 4 4 0¦ 0 3¦ 3 1 1 2 2 4 4 0¦ 1 2 2 3 3 4 4 0 0 1 2 1 1 3 3 4 4 0 0 2 3 1 1 2 2 4 4 0 0 3 2¦ 3 4 0 1¦ 1¦ 3 4 0 2¦ 1¦ 2 4 0 3¦ 3-----4-----0-----1-----2 3-----4-----0-----2-----1 2-----4-----0-----3-----1

16

73/ 97/ 121/ 0-----3-----2-----1-----4 1-----0-----2-----3-----4 1-----2-----0-----3-----4 ¦4 0 3 2 ¦1 ¦0 2 3 4 ¦1 ¦2 0 3 4 ¦1 3 1 2 4 1 0 4 3 0 2 0 2 2 3 3 4 4 1 1 0 2 0 0 3 3 4 4 1 1 2 ¦0 2 3 1 2 4 1 0 ¦4 3 ¦2 3 3 4 4 1 1 0 ¦0 2 ¦0 3 3 4 4 1 1 2 ¦2 0 2 4 3-1-0-2-4-3-1-0-2-4-3-1-0 2 3 4-3-4-1-4-1-0-1-0-2-0-2-3 0 3 4-3-4-1-4-1-2-1-2-0-2-0-3 ¦3 1¦ 2 4 1 0 4 3 ¦0 2¦ ¦3 4¦ 4 1 1 0 0 2 ¦2 3¦ ¦3 4¦ 4 1 1 2 2 0 ¦0 3¦ 1 0 2 4 3 1 0 2 4 3 1 0 2 4 3 3 4 1 4 1 0 1 0 2 0 2 3 2 3 4 3 4 1 4 1 2 1 2 0 2 0 3 0 3 4 ¦2 4¦ 1 0 4 3 0 2 ¦3 1¦ ¦4 1¦ 1 0 0 2 2 3 ¦3 4¦ ¦4 1¦ 1 2 2 0 0 3 ¦3 4¦ 4-3-1-0-2-4-3-1-0-2-4-3-1 0 2 4-1-0-1-0-2-0-2-3-2-3-4-3 4 1 4-1-2-1-2-0-2-0-3-0-3-4-3 4 1 1 0¦ 4 3 0 2 3 1 2 4¦ 1 0¦ 0 2 2 3 3 4 4 1¦ 1 2¦ 2 0 0 3 3 4 4 1¦ 2 4 1 0 4 3 0 2 3 1 0 2 2 3 3 4 4 1 1 0 2 0 0 3 3 4 4 1 1 2 3¦ 2 1 4 0¦ 2¦ 3 4 1 0¦ 0¦ 3 4 1 2¦ 0-----3-----2-----1-----4 3-----4-----1-----0-----2 3-----4-----1-----2-----0

145/ 169/ 193/ 1-----3-----0-----2-----4 1-----3-----2-----0-----4 2-----0-----1-----3-----4 ¦3 0 2 4 ¦1 ¦4 1 3 2 ¦0 ¦0 1 3 4 ¦2 3 0 0 2 2 4 4 1 1 3 3 0 2 4 0 1 4 3 1 2 0 1 1 3 3 4 4 2 2 0 ¦0 2 2 4 4 1 1 3 ¦3 0 ¦1 2 3 0 2 4 0 1 ¦4 3 ¦1 3 3 4 4 2 2 0 ¦0 1 0 2 4-2-4-1-4-1-3-1-3-0-3-0-2 2 4 3-0-1-2-4-3-0-1-2-4-3-0-1 1 3 4-3-4-2-4-2-0-2-0-1-0-1-3 ¦2 4¦ 4 1 1 3 3 0 ¦0 2¦ ¦3 0¦ 2 4 0 1 4 3 ¦1 2¦ ¦3 4¦ 4 2 2 0 0 1 ¦1 3¦ 2 4 1 4 1 3 1 3 0 3 0 2 0 2 4 0 1 2 4 3 0 1 2 4 3 0 1 2 4 3 3 4 2 4 2 0 2 0 1 0 1 3 1 3 4 ¦4 1¦ 1 3 3 0 0 2 ¦2 4¦ ¦2 4¦ 0 1 4 3 1 2 ¦3 0¦ ¦4 2¦ 2 0 0 1 1 3 ¦3 4¦ 4-1-3-1-3-0-3-0-2-0-2-4-2 4 1 4-3-0-1-2-4-3-0-1-2-4-3-0 1 2 4-2-0-2-0-1-0-1-3-1-3-4-3 4 2 1 3¦ 3 0 0 2 2 4 4 1¦ 0 1¦ 4 3 1 2 3 0 2 4¦ 2 0¦ 0 1 1 3 3 4 4 2¦ 3 0 0 2 2 4 4 1 1 3 2 4 0 1 4 3 1 2 3 0 0 1 1 3 3 4 4 2 2 0 0¦ 2 4 1 3¦ 3¦ 2 0 4 1¦ 1¦ 3 4 2 0¦ 2-----4-----1-----3-----0 1-----3-----2-----0-----4 3-----4-----2-----0-----1

217/ 241/ 265/ 2-----1-----0-----3-----4 2-----3-----0-----1-----4 2-----3-----1-----0-----4 ¦1 0 3 4 ¦2 ¦3 0 1 4 ¦2 ¦4 2 3 1 ¦0 1 0 0 3 3 4 4 2 2 1 3 0 0 1 1 4 4 2 2 3 3 0 1 4 0 2 4 3 2 1 ¦0 3 3 4 4 2 2 1 ¦1 0 ¦0 1 1 4 4 2 2 3 ¦3 0 ¦2 1 3 0 1 4 0 2 ¦4 3 0 3 4-3-4-2-4-2-1-2-1-0-1-0-3 0 1 4-1-4-2-4-2-3-2-3-0-3-0-1 1 4 3-0-2-1-4-3-0-2-1-4-3-0-2 ¦3 4¦ 4 2 2 1 1 0 ¦0 3¦ ¦1 4¦ 4 2 2 3 3 0 ¦0 1¦ ¦3 0¦ 1 4 0 2 4 3 ¦2 1¦ 3 4 2 4 2 1 2 1 0 1 0 3 0 3 4 1 4 2 4 2 3 2 3 0 3 0 1 0 1 4 0 2 1 4 3 0 2 1 4 3 0 2 1 4 3 ¦4 2¦ 2 1 1 0 0 3 ¦3 4¦ ¦4 2¦ 2 3 3 0 0 1 ¦1 4¦ ¦1 4¦ 0 2 4 3 2 1 ¦3 0¦ 4-2-1-2-1-0-1-0-3-0-3-4-3 4 2 4-2-3-2-3-0-3-0-1-0-1-4-1 4 2 4-3-0-2-1-4-3-0-2-1-4-3-0 2 1 2 1¦ 1 0 0 3 3 4 4 2¦ 2 3¦ 3 0 0 1 1 4 4 2¦ 0 2¦ 4 3 2 1 3 0 1 4¦ 1 0 0 3 3 4 4 2 2 1 3 0 0 1 1 4 4 2 2 3 1 4 0 2 4 3 2 1 3 0 0¦ 3 4 2 1¦ 0¦ 1 4 2 3¦ 3¦ 1 0 4 2¦ 3-----4-----2-----1-----0 1-----4-----2-----3-----0 2-----3-----1-----0-----4

289/ 313/ 337/ 3-----0-----1-----2-----4 3-----1-----0-----2-----4 3-----2-----0-----1-----4 ¦0 1 2 4 ¦3 ¦1 0 2 4 ¦3 ¦2 0 1 4 ¦3 0 1 1 2 2 4 4 3 3 0 1 0 0 2 2 4 4 3 3 1 2 0 0 1 1 4 4 3 3 2 ¦1 2 2 4 4 3 3 0 ¦0 1 ¦0 2 2 4 4 3 3 1 ¦1 0 ¦0 1 1 4 4 3 3 2 ¦2 0 1 2 4-2-4-3-4-3-0-3-0-1-0-1-2 0 2 4-2-4-3-4-3-1-3-1-0-1-0-2 0 1 4-1-4-3-4-3-2-3-2-0-2-0-1 ¦2 4¦ 4 3 3 0 0 1 ¦1 2¦ ¦2 4¦ 4 3 3 1 1 0 ¦0 2¦ ¦1 4¦ 4 3 3 2 2 0 ¦0 1¦ 2 4 3 4 3 0 3 0 1 0 1 2 1 2 4 2 4 3 4 3 1 3 1 0 1 0 2 0 2 4 1 4 3 4 3 2 3 2 0 2 0 1 0 1 4 ¦4 3¦ 3 0 0 1 1 2 ¦2 4¦ ¦4 3¦ 3 1 1 0 0 2 ¦2 4¦ ¦4 3¦ 3 2 2 0 0 1 ¦1 4¦ 4-3-0-3-0-1-0-1-2-1-2-4-2 4 3 4-3-1-3-1-0-1-0-2-0-2-4-2 4 3 4-3-2-3-2-0-2-0-1-0-1-4-1 4 3 3 0¦ 0 1 1 2 2 4 4 3¦ 3 1¦ 1 0 0 2 2 4 4 3¦ 3 2¦ 2 0 0 1 1 4 4 3¦ 0 1 1 2 2 4 4 3 3 0 1 0 0 2 2 4 4 3 3 1 2 0 0 1 1 4 4 3 3 2 1¦ 2 4 3 0¦ 0¦ 2 4 3 1¦ 0¦ 1 4 3 2¦ 2-----4-----3-----0-----1 2-----4-----3-----1-----0 1-----4-----3-----2-----0

361/ 385/ 409/ 3-----2-----1-----0-----4 4-----0-----1-----2-----3 4-----1-----0-----2-----3 ¦4 3 2 1 ¦0 ¦0 1 2 3 ¦4 ¦1 0 2 3 ¦4 2 0 1 4 0 3 4 2 3 1 0 1 1 2 2 3 3 4 4 0 1 0 0 2 2 3 3 4 4 1 ¦3 1 2 0 1 4 0 3 ¦4 2 ¦1 2 2 3 3 4 4 0 ¦0 1 ¦0 2 2 3 3 4 4 1 ¦1 0 1 4 2-0-3-1-4-2-0-3-1-4-2-0-3 1 2 3-2-3-4-3-4-0-4-0-1-0-1-2 0 2 3-2-3-4-3-4-1-4-1-0-1-0-2 ¦2 0¦ 1 4 0 3 4 2 ¦3 1¦ ¦2 3¦ 3 4 4 0 0 1 ¦1 2¦ ¦2 3¦ 3 4 4 1 1 0 ¦0 2¦ 0 3 1 4 2 0 3 1 4 2 0 3 1 4 2 2 3 4 3 4 0 4 0 1 0 1 2 1 2 3 2 3 4 3 4 1 4 1 0 1 0 2 0 2 3 ¦1 4¦ 0 3 4 2 3 1 ¦2 0¦ ¦3 4¦ 4 0 0 1 1 2 ¦2 3¦ ¦3 4¦ 4 1 1 0 0 2 ¦2 3¦ 4-2-0-3-1-4-2-0-3-1-4-2-0 3 1 3-4-0-4-0-1-0-1-2-1-2-3-2 3 4 3-4-1-4-1-0-1-0-2-0-2-3-2 3 4 0 3¦ 4 2 3 1 2 0 1 4¦ 4 0¦ 0 1 1 2 2 3 3 4¦ 4 1¦ 1 0 0 2 2 3 3 4¦ 1 4 0 3 4 2 3 1 2 0 0 1 1 2 2 3 3 4 4 0 1 0 0 2 2 3 3 4 4 1 2¦ 1 0 4 3¦ 1¦ 2 3 4 0¦ 0¦ 2 3 4 1¦ 3-----2-----1-----0-----4 2-----3-----4-----0-----1 2-----3-----4-----1-----0

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433/ 457/ 4-----2-----0-----1-----3 4-----2-----1-----0-----3 ¦2 0 1 3 ¦4 ¦3 4 2 1 ¦0 2 0 0 1 1 3 3 4 4 2 2 0 1 3 0 4 3 2 4 1 ¦0 1 1 3 3 4 4 2 ¦2 0 ¦4 1 2 0 1 3 0 4 ¦3 2 0 1 3-1-3-4-3-4-2-4-2-0-2-0-1 1 3 2-0-4-1-3-2-0-4-1-3-2-0-4 ¦1 3¦ 3 4 4 2 2 0 ¦0 1¦ ¦2 0¦ 1 3 0 4 3 2 ¦4 1¦ 1 3 4 3 4 2 4 2 0 2 0 1 0 1 3 0 4 1 3 2 0 4 1 3 2 0 4 1 3 2 ¦3 4¦ 4 2 2 0 0 1 ¦1 3¦ ¦1 3¦ 0 4 3 2 4 1 ¦2 0¦ 3-4-2-4-2-0-2-0-1-0-1-3-1 3 4 3-2-0-4-1-3-2-0-4-1-3-2-0 4 1 4 2¦ 2 0 0 1 1 3 3 4¦ 0 4¦ 3 2 4 1 2 0 1 3¦ 2 0 0 1 1 3 3 4 4 2 1 3 0 4 3 2 4 1 2 0 0¦ 1 3 4 2¦ 2¦ 1 0 3 4¦ 1-----3-----4-----2-----0 4-----2-----1-----0-----3

[Total Count of Layer Units = 480]

7. How to Combine Latin Units and Compose "Complete Euler Cubes" of Order 5: At first we combine any three units picked up, calculate the positional data one after another according to the next equation and compose our final object as follows. 1/H 2/M 3/L 0-----1-----2-----3-----4 0-----4-----3-----2-----1 0-----1-----2-----3-----4 ¦1 2 3 4 ¦0 ¦1 0 4 3 ¦2 ¦1 2 3 4 ¦0 1 2 2 3 3 4 4 0 0 1 1 2 0 1 4 0 3 4 2 3 4 2 0 3 1 4 2 0 3 1 ¦2 3 3 4 4 0 0 1 ¦1 2 ¦2 3 1 2 0 1 4 0 ¦3 4 ¦0 3 1 4 2 0 3 1 ¦4 2 2 3 4-3-4-0-4-0-1-0-1-2-1-2-3 2 3 4-1-2-3-0-1-2-4-0-1-3-4-0 3 1 4-4-2-0-0-3-1-1-4-2-2-0-3 ¦3 4¦ 4 0 0 1 1 2 ¦2 3¦ ¦3 4¦ 2 3 1 2 0 1 ¦4 0¦ ¦4 2¦ 0 3 1 4 2 0 ¦3 1¦ 3 4 0 4 0 1 0 1 2 1 2 3 2 3 4 3 4 0 2 3 4 1 2 3 0 1 2 4 0 1 2 0 3 3 1 4 4 2 0 0 3 1 1 4 2 ¦4 0¦ 0 1 1 2 2 3 ¦3 4¦ ¦4 0¦ 3 4 2 3 1 2 ¦0 1¦ ¦3 1¦ 4 2 0 3 1 4 ¦2 0¦ 4-0-1-0-1-2-1-2-3-2-3-4-3 4 0 4-0-1-3-4-0-2-3-4-1-2-3-0 1 2 1-4-2-2-0-3-3-1-4-4-2-0-0 3 1 0 1¦ 1 2 2 3 3 4 4 0¦ 0 1¦ 4 0 3 4 2 3 1 2¦ 2 0¦ 3 1 4 2 0 3 1 4¦ 1 2 2 3 3 4 4 0 0 1 1 2 0 1 4 0 3 4 2 3 3 1 4 2 0 3 1 4 2 0 2¦ 3 4 0 1¦ 2¦ 1 0 4 3¦ 4¦ 0 1 2 3¦ 3-----4-----0-----1-----2 3-----2-----1-----0-----4 0-----1-----2-----3-----4

1/H 2/M 3/L 1# 1-------------- 47-------------- 68-------------- 89--------------110 ¦ 32 53 99 120 ¦ 11 35 63 51 84 97 105 118 21 14 42 ¦ 61 94 82 115 103 6 24 27 ¦ 45 73 64 92 125-- 85---113--- 16--101--- 9--- 37-- 22--- 30--- 58-- 43--- 71--- 79 ¦ 95 123 ¦ 111 19 7 40 28 56 ¦ 74 77 ¦ 93 121 4 114 17 50 10 38 66 26 59 87 72 80 108 ¦124 2 ¦ 20 48 36 69 57 90 ¦ 78 106 ¦ 122--- 5--- 33-- 18--- 46--- 54-- 39--- 67---100-- 60--- 88---116-- 76 109 12 3 31 ¦ 49 52 70 98 86 119 107 15 ¦ 34 62 55 83 96 104 117 25 13 41 65 ¦ 81 102 23 44 ¦ 91--------------112-------------- 8-------------- 29-------------- 75

n_of_High, Middle, and Low N_of_Object_Cube

(nH x 5 + nM) x 5 + nL + 1 = nH x 25 + nM x 5 + nL + 1 = NC

0H x 25 + 0M x 5 + 0L + 1 = 0+0+0+1 = 1 ...n1C;

1H x 25 + 1M x 5 + 1L + 1 = 25+5+1+1 = 32 ...n2C; 2H x 25 + 2M x 5 + 2L + 1 = 50+10+2+1 = 63 ...n3C; 3H x 25 + 3M x 5 + 3L + 1 = 75+15+3+1 = 94 ...n4C; 4H x 25 + 4M x 5 + 4L + 1 = 100+20+4+1 = 125 ...n5C; 1H x 25 + 1M x 5 + 4L + 1 = 25+5+4+1 = 35 ...n6C; . . . . 1H x 25 + 4M x 5 + 1L + 1 = 25+20+1+1 = 47 ...n26C; . . . .

Why do we have to add 1 to the end of every calculation above? Because we want to

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match the two styles of diagram expression between Classical and Mathematical. The classical expression uses a series of natural numbers from 1 to 125, while the other mathematical one uses a series of integers from 0 to 124. And our Positional Number System of Base 5 also uses integers from 0 to 4. We must also remember we should make up our object of a series of numbers from 1 to 125 using each one strictly once. In the expression of Positional (5-th Increment) Number System, we have to use all {000(N5i), 001(N5i), 002(N5i), 003(N5i), 004(N5i), 100(N5i), 101(N5i), ... ,434(N5i), 440(N5i), 441(N5i), 442(N5i), 443(N5i), 444(N5i)} taking each number strictly once. I say, this is one of the most important rules we should always obey. Neither duplication nor lack of any number must be found there. Tests are needed for the second step. Examine every actual result one after another to find out whether it is right or wrong. Check if it surely obeys the first definition. And find if it has neither duplication nor lack of any number in it. Every combination of Latin Units does not always make the correct solution, you should know. But it often makes a wrong solution.

When you combine any two units which have too little difference or too much difference between them, you will surely have a wrong solution. It has a lot of duplication or lack of same numbers in it. Only selected combinations could compose correct solutions. Examine this two times: both before the calculation and after that. We have to make a reference table that has 'resemblance' data between any two units before all, measuring how many variables have the same values or the complimentary ones. You can come this table before you combine any two units, and refer to it about their compatibility. After the calculation, you also have to examine your composition to find if it really has duplication or lack of any number. Since we must have the set of Standard Solutions of our object cubes, we have to eliminate all the reflected patterns or the rotated ones from the Primitive Solutions. Standard solutions are found among primitive in 1/48 necessarily in any case of magic cubes. What can we do before or after our calculations? Then we have to prepare two sets of Inequality Conditions defining the relations among these critical variables: n1<n5; n1<n21; n1<n25; n1<n101; n1<n105; n1<n121; n1<n125; n2<n6; n6<n26; 8. How to dictate our program for our 'C.E.C.' of Order5 Let me list out the second part of my Program, only a part. /**/ long int cnt; short cnt0, cnt1, cnt2, cnt3; short u1, u2, u3; short nm[126], flg[126]; short bc[176][5]; short d[4][129]; short tluh[481]; short tlu[481][126]; char mtc[481][481]; /**/ /* Main Program */ int main(){ short m,n; printf("\n** 'Latin Cubes' of Order 5 for the Pan-triagonal Type: **\n"); printf("** by Kanji Setsuda on June 12, 2010 with Xcode 3.1.2 **\n"); cnt=0; cnt0=0; cnt1=0; cnt3=0; for(n=0;n<126;n++){nm[n]=0;}

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for(m=1;m<176;m++){for(n=0;n<5;n++){bc[m][n]=0;}} for(n=0;n<481;n++){tluh[n]=0;} stp01(); mkreftbl(); printf("\n*** 'Complete Euler Cubes' of Order 5 for the Pan-triagonal Type: ***\n"); printf("** Made by Kanji Setsuda on June 12, 2010 with MacOSX & Xcode 3.1.2 **\n"); cnt=0; cnt3=0; cmbcmp(); printf(" [Total Count of Standard Solutions = %d] OK!\n",cnt); return 0; } /* ...... */ /**/ /* Combine and Compose */ void cmbcmp(){ short n,d,lu,md,fc; lu=480; md=25; //'md' means the Matching Data for the best Compatibility. for(u1=0;u1<lu;u1++){ if(tluh[u1]>0){cnt2=0; for(u2=0;u2<lu;u2++){ if(mtc[u2][u1]==md){ //cnt2=0; for(u3=0;u3<lu;u3++){ if((mtc[u3][u1]==md)&&(mtc[u3][u2]==md)){ //cnt2=0; for(n=1;n<126;n++){flg[n]=0;} for(n=1;n<126;n++){ d=tlu[u1][n]*25+tlu[u2][n]*5+tlu[u3][n]+1; nm[n]=d; flg[d]++; } fc=0; for(n=1;n<126;n++){ if(flg[n]==1){fc++;}else{break;} } if(fc==125){d=nm[1]; if((d<nm[5])&&(d<nm[21])&&(d<nm[25])&&(d<nm[101])&&(d<nm[105])){ if((d<nm[121])&&(d<nm[125])&&(nm[2]<nm[6])&&(nm[6]<nm[26])){ pransl(); }}} } } } }} } } /**/ void pransl(void){ short m,n,l; cnt++; cnt2++; if(cnt2==1){ d[0][0]=u1+1; d[1][0]=u2+1; d[2][0]=u3+1; for(m=0;m<3;m++){l=d[m][0]-1; for(n=1;n<126;n++){d[m][n]=tlu[l][n];} } prdlu(4); prptl(); } } /**/ /* Print the Result */ void prptl(void){ printf("%10d/H%5d/M%5d/L%45d#\n",u1+1,u2+1,u3+1,cnt); printf("%4d--------------%3d--------------%3d--------------%3d--------------%3d\n", nm[1],nm[26],nm[51],nm[76],nm[101]); printf(" |%3d%17d%17d%17d |%3d\n", nm[2],nm[27],nm[52],nm[77],nm[102]); printf("%4d%6d %3d%6d %3d%6d %3d%6d %3d%6d\n",

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nm[6],nm[3],nm[31],nm[28],nm[56],nm[53],nm[81],nm[78],nm[106],nm[103]); printf(" |%3d%6d %3d%6d %3d%6d %3d%6d |%3d%6d\n", nm[7],nm[4],nm[32],nm[29],nm[57],nm[54],nm[82],nm[79],nm[107],nm[104]); printf("%4d%6d%6d--%3d---%3d---%3d--%3d---%3d---%3d--%3d---%3d---%3d--%3d---%3d---%3d\n", nm[11],nm[8],nm[5],nm[36],nm[33],nm[30],nm[61],nm[58],nm[55], nm[86],nm[83],nm[80],nm[111],nm[108],nm[105]); printf(" |%3d%6d | %3d%6d %3d%6d %3d%6d |%3d%6d |\n", nm[12],nm[9],nm[37],nm[34],nm[62],nm[59],nm[87],nm[84],nm[112],nm[109]); printf("%4d%6d%6d%5d%6d%6d%5d%6d%6d%5d%6d%6d%5d%6d%6d\n", nm[16],nm[13],nm[10],nm[41],nm[38],nm[35],nm[66],nm[63],nm[60], nm[91],nm[88],nm[85],nm[116],nm[113],nm[110]); printf(" |%3d%6d | %3d%6d %3d%6d %3d%6d |%3d%6d |\n", nm[17],nm[14],nm[42],nm[39],nm[67],nm[64],nm[92],nm[89],nm[117],nm[114]); printf("%4d---%3d---%3d--%3d---%3d---%3d--%3d---%3d---%3d--%3d---%3d---%3d--%3d%6d%6d\n", nm[21],nm[18],nm[15],nm[46],nm[43],nm[40],nm[71],nm[68],nm[65], nm[96],nm[93],nm[90],nm[121],nm[118],nm[115]); printf("%7d%6d | %3d%6d %3d%6d %3d%6d %3d%6d |\n", nm[22],nm[19],nm[47],nm[44],nm[72],nm[69],nm[97],nm[94],nm[122],nm[119]); printf("%10d%6d %3d%6d %3d%6d %3d%6d %3d%6d\n", nm[23],nm[20],nm[48],nm[45],nm[73],nm[70],nm[98],nm[95],nm[123],nm[120]); printf("%13d |%14d%17d%17d%17d |\n", nm[24],nm[49],nm[74],nm[99],nm[124]); printf("%16d--------------%3d--------------%3d--------------%3d--------------%3d\n", nm[25],nm[50],nm[75],nm[100],nm[125]); } /**/ void prdlu(short x){ short i; if(x>3){printf("%24d/H%29d/M%29d/L\n",d[0][0],d[1][0],d[2][0]); x=3;} else if(x==3){printf("%25d/%30d/%30d/\n",d[0][0],d[1][0],d[2][0]);} else if(x==2){printf("%25d/%30d/\n",d[0][0],d[1][0]);} else{printf("%25d/\n",d[0][0]);} for(i=0;i<x;i++){ printf("%2d-----%d-----%d-----%d-----%d",d[i][1],d[i][26],d[i][51],d[i][76],d[i][101]); if(i<2){printf(" ");}} printf("\n"); for(i=0;i<x;i++){ printf(" |%d%6d%6d%6d |%d",d[i][2],d[i][27],d[i][52],d[i][77],d[i][102]); if(i<2){printf(" ");}} printf("\n"); for(i=0;i<x;i++){ printf("%2d%2d%4d%2d%4d%2d%4d%2d%4d%2d", d[i][6],d[i][3],d[i][31],d[i][28],d[i][56],d[i][53],d[i][81],d[i][78],d[i][106],d[i][103]); if(i<2){printf(" ");}} printf("\n"); for(i=0;i<x;i++){ printf(" |%d%2d%4d%2d%4d%2d%4d%2d |%d%2d", d[i][7],d[i][4],d[i][32],d[i][29],d[i][57],d[i][54],d[i][82],d[i][79],d[i][107],d[i][104]); if(i<2){printf(" ");}} printf("\n"); for(i=0;i<x;i++){ printf("%2d%2d%2d-%d-%d-%d-%d-%d-%d-%d-%d-%d-%d-%d-%d", d[i][11],d[i][8],d[i][5],d[i][36],d[i][33],d[i][30],d[i][61],d[i][58],d[i][55], d[i][86],d[i][83],d[i][80],d[i][111],d[i][108],d[i][105]); if(i<2){printf(" ");}} printf("\n"); for(i=0;i<x;i++){ printf(" |%d%2d|%3d%2d%4d%2d%4d%2d |%d%2d|", d[i][12],d[i][9],d[i][37],d[i][34],d[i][62],d[i][59],d[i][87],d[i][84],d[i][112],d[i][109]); if(i<2){printf(" ");}} printf("\n"); for(i=0;i<x;i++){ printf("%2d%2d%2d%2d%2d%2d%2d%2d%2d%2d%2d%2d%2d%2d%2d", d[i][16],d[i][13],d[i][10],d[i][41],d[i][38],d[i][35],d[i][66],d[i][63],d[i][60], d[i][91],d[i][88],d[i][85],d[i][116],d[i][113],d[i][110]);

21

if(i<2){printf(" ");}} printf("\n"); for(i=0;i<x;i++){ printf(" |%d%2d|%3d%2d%4d%2d%4d%2d |%d%2d|", d[i][17],d[i][14],d[i][42],d[i][39],d[i][67],d[i][64],d[i][92],d[i][89],d[i][117],d[i][114]); if(i<2){printf(" ");}} printf("\n"); for(i=0;i<x;i++){ printf("%2d-%d-%d-%d-%d-%d-%d-%d-%d-%d-%d-%d-%d%2d%2d", d[i][21],d[i][18],d[i][15],d[i][46],d[i][43],d[i][40],d[i][71],d[i][68],d[i][65], d[i][96],d[i][93],d[i][90],d[i][121],d[i][118],d[i][115]); if(i<2){printf(" ");}} printf("\n"); for(i=0;i<x;i++){ printf(" %d%2d|%3d%2d%4d%2d%4d%2d%4d%2d|", d[i][22],d[i][19],d[i][47],d[i][44],d[i][72],d[i][69],d[i][97],d[i][94],d[i][122],d[i][119]); if(i<2){printf(" ");}} printf("\n"); for(i=0;i<x;i++){ printf(" %2d%2d%4d%2d%4d%2d%4d%2d%4d%2d", d[i][23],d[i][20],d[i][48],d[i][45],d[i][73],d[i][70],d[i][98],d[i][95],d[i][123],d[i][120]); if(i<2){printf(" ");}} printf("\n"); for(i=0;i<x;i++){ printf("%5d|%5d%6d%6d%6d|", d[i][24],d[i][49],d[i][74],d[i][99],d[i][124]); if(i<2){printf(" ");}} printf("\n"); for(i=0;i<x;i++){ printf("%6d-----%d-----%d-----%d-----%d", d[i][25],d[i][50],d[i][75],d[i][100],d[i][125]); if(i<2){printf(" ");}} printf("\n"); } /**/ 9. What I have got as for the result of my hardest computation: Let me list out the recent result of my compositions, though only a part. I have not yet sorted to rearrange the listing order of solutions to any smarter style. *** 'Complete Euler Cubes' of Order 5 for the Pan-triagonal Type *** ** Made by Kanji Setsuda on June 12, 2010 with MacOSX & Xcode 3.1.2 ** 1/H 2/M 3/L 0-----1-----2-----3-----4 0-----4-----3-----2-----1 0-----1-----2-----3-----4 ¦1 2 3 4 ¦0 ¦1 0 4 3 ¦2 ¦1 2 3 4 ¦0 1 2 2 3 3 4 4 0 0 1 1 2 0 1 4 0 3 4 2 3 4 2 0 3 1 4 2 0 3 1 ¦2 3 3 4 4 0 0 1 ¦1 2 ¦2 3 1 2 0 1 4 0 ¦3 4 ¦0 3 1 4 2 0 3 1 ¦4 2 2 3 4-3-4-0-4-0-1-0-1-2-1-2-3 2 3 4-1-2-3-0-1-2-4-0-1-3-4-0 3 1 4-4-2-0-0-3-1-1-4-2-2-0-3 ¦3 4¦ 4 0 0 1 1 2 ¦2 3¦ ¦3 4¦ 2 3 1 2 0 1 ¦4 0¦ ¦4 2¦ 0 3 1 4 2 0 ¦3 1¦ 3 4 0 4 0 1 0 1 2 1 2 3 2 3 4 3 4 0 2 3 4 1 2 3 0 1 2 4 0 1 2 0 3 3 1 4 4 2 0 0 3 1 1 4 2 ¦4 0¦ 0 1 1 2 2 3 ¦3 4¦ ¦4 0¦ 3 4 2 3 1 2 ¦0 1¦ ¦3 1¦ 4 2 0 3 1 4 ¦2 0¦ 4-0-1-0-1-2-1-2-3-2-3-4-3 4 0 4-0-1-3-4-0-2-3-4-1-2-3-0 1 2 1-4-2-2-0-3-3-1-4-4-2-0-0 3 1 0 1¦ 1 2 2 3 3 4 4 0¦ 0 1¦ 4 0 3 4 2 3 1 2¦ 2 0¦ 3 1 4 2 0 3 1 4¦ 1 2 2 3 3 4 4 0 0 1 1 2 0 1 4 0 3 4 2 3 3 1 4 2 0 3 1 4 2 0 2¦ 3 4 0 1¦ 2¦ 1 0 4 3¦ 4¦ 0 1 2 3¦ 3-----4-----0-----1-----2 3-----2-----1-----0-----4 0-----1-----2-----3-----4

1/H 2/M 3/L 1# 1-------------- 47-------------- 68-------------- 89--------------110 ¦ 32 53 99 120 ¦ 11 35 63 51 84 97 105 118 21 14 42 ¦ 61 94 82 115 103 6 24 27 ¦ 45 73 64 92 125-- 85---113--- 16--101--- 9--- 37-- 22--- 30--- 58-- 43--- 71--- 79 ¦ 95 123 ¦ 111 19 7 40 28 56 ¦ 74 77 ¦

22

93 121 4 114 17 50 10 38 66 26 59 87 72 80 108 ¦124 2 ¦ 20 48 36 69 57 90 ¦ 78 106 ¦ 122--- 5--- 33-- 18--- 46--- 54-- 39--- 67---100-- 60--- 88---116-- 76 109 12 3 31 ¦ 49 52 70 98 86 119 107 15 ¦ 34 62 55 83 96 104 117 25 13 41 65 ¦ 81 102 23 44 ¦ 91--------------112-------------- 8-------------- 29-------------- 75

1/H 2/M 4/L 0-----1-----2-----3-----4 0-----4-----3-----2-----1 0-----4-----3-----2-----1 ¦1 2 3 4 ¦0 ¦1 0 4 3 ¦2 ¦1 0 4 3 ¦2 1 2 2 3 3 4 4 0 0 1 1 2 0 1 4 0 3 4 2 3 4 2 3 1 2 0 1 4 0 3 ¦2 3 3 4 4 0 0 1 ¦1 2 ¦2 3 1 2 0 1 4 0 ¦3 4 ¦0 3 4 2 3 1 2 0 ¦1 4 2 3 4-3-4-0-4-0-1-0-1-2-1-2-3 2 3 4-1-2-3-0-1-2-4-0-1-3-4-0 3 1 4-2-0-3-1-4-2-0-3-1-4-2-0 ¦3 4¦ 4 0 0 1 1 2 ¦2 3¦ ¦3 4¦ 2 3 1 2 0 1 ¦4 0¦ ¦4 2¦ 3 1 2 0 1 4 ¦0 3¦ 3 4 0 4 0 1 0 1 2 1 2 3 2 3 4 3 4 0 2 3 4 1 2 3 0 1 2 4 0 1 2 0 3 1 4 2 0 3 1 4 2 0 3 1 4 ¦4 0¦ 0 1 1 2 2 3 ¦3 4¦ ¦4 0¦ 3 4 2 3 1 2 ¦0 1¦ ¦3 1¦ 2 0 1 4 0 3 ¦4 2¦ 4-0-1-0-1-2-1-2-3-2-3-4-3 4 0 4-0-1-3-4-0-2-3-4-1-2-3-0 1 2 1-4-2-0-3-1-4-2-0-3-1-4-2 0 3 0 1¦ 1 2 2 3 3 4 4 0¦ 0 1¦ 4 0 3 4 2 3 1 2¦ 2 0¦ 1 4 0 3 4 2 3 1¦ 1 2 2 3 3 4 4 0 0 1 1 2 0 1 4 0 3 4 2 3 3 1 2 0 1 4 0 3 4 2 2¦ 3 4 0 1¦ 2¦ 1 0 4 3¦ 4¦ 3 2 1 0¦ 3-----4-----0-----1-----2 3-----2-----1-----0-----4 0-----4-----3-----2-----1

1/H 2/M 4/L 2# 1-------------- 50-------------- 69-------------- 88--------------107 ¦ 32 51 100 119 ¦ 13 35 63 54 82 98 101 117 25 11 44 ¦ 61 94 85 113 104 7 23 26 ¦ 42 75 64 92 125-- 83---111--- 19--102--- 10--- 38-- 21--- 29--- 57-- 45--- 73--- 76 ¦ 95 123 ¦ 114 17 8 36 27 60 ¦ 71 79 ¦ 93 121 4 112 20 48 6 39 67 30 58 86 74 77 110 ¦124 2 ¦ 18 46 37 70 56 89 ¦ 80 108 ¦ 122--- 5--- 33-- 16--- 49--- 52-- 40--- 68--- 96-- 59--- 87---120-- 78 106 14 3 31 ¦ 47 55 66 99 90 118 109 12 ¦ 34 62 53 81 97 105 116 24 15 43 65 ¦ 84 103 22 41 ¦ 91--------------115-------------- 9-------------- 28-------------- 72

1/H 2/M 7/L 0-----1-----2-----3-----4 0-----4-----3-----2-----1 0-----1-----2-----4-----3 ¦1 2 3 4 ¦0 ¦1 0 4 3 ¦2 ¦1 2 4 3 ¦0 1 2 2 3 3 4 4 0 0 1 1 2 0 1 4 0 3 4 2 3 3 2 0 4 1 3 2 0 4 1 ¦2 3 3 4 4 0 0 1 ¦1 2 ¦2 3 1 2 0 1 4 0 ¦3 4 ¦0 4 1 3 2 0 4 1 ¦3 2 2 3 4-3-4-0-4-0-1-0-1-2-1-2-3 2 3 4-1-2-3-0-1-2-4-0-1-3-4-0 4 1 3-3-2-0-0-4-1-1-3-2-2-0-4 ¦3 4¦ 4 0 0 1 1 2 ¦2 3¦ ¦3 4¦ 2 3 1 2 0 1 ¦4 0¦ ¦3 2¦ 0 4 1 3 2 0 ¦4 1¦ 3 4 0 4 0 1 0 1 2 1 2 3 2 3 4 3 4 0 2 3 4 1 2 3 0 1 2 4 0 1 2 0 4 4 1 3 3 2 0 0 4 1 1 3 2 ¦4 0¦ 0 1 1 2 2 3 ¦3 4¦ ¦4 0¦ 3 4 2 3 1 2 ¦0 1¦ ¦4 1¦ 3 2 0 4 1 3 ¦2 0¦ 4-0-1-0-1-2-1-2-3-2-3-4-3 4 0 4-0-1-3-4-0-2-3-4-1-2-3-0 1 2 1-3-2-2-0-4-4-1-3-3-2-0-0 4 1 0 1¦ 1 2 2 3 3 4 4 0¦ 0 1¦ 4 0 3 4 2 3 1 2¦ 2 0¦ 4 1 3 2 0 4 1 3¦ 1 2 2 3 3 4 4 0 0 1 1 2 0 1 4 0 3 4 2 3 4 1 3 2 0 4 1 3 2 0 2¦ 3 4 0 1¦ 2¦ 1 0 4 3¦ 3¦ 0 1 2 4¦ 3-----4-----0-----1-----2 3-----2-----1-----0-----4 0-----1-----2-----4-----3

1/H 2/M 7/L 3# 1-------------- 47-------------- 68-------------- 90--------------109 ¦ 32 53 100 119 ¦ 11 34 63 51 85 97 104 118 21 15 42 ¦ 61 95 82 114 103 6 25 27 ¦ 44 73 65 92 124-- 84---113--- 16--101--- 10--- 37-- 22--- 29--- 58-- 43--- 71--- 80 ¦ 94 123 ¦ 111 20 7 39 28 56 ¦ 75 77 ¦ 93 121 5 115 17 49 9 38 66 26 60 87 72 79 108 ¦125 2 ¦ 19 48 36 70 57 89 ¦ 78 106 ¦ 122--- 4--- 33-- 18--- 46--- 55-- 40--- 67--- 99-- 59--- 88---116-- 76 110 12 3 31 ¦ 50 52 69 98 86 120 107 14 ¦ 35 62 54 83 96 105 117 24 13 41 64 ¦ 81 102 23 45 ¦ 91--------------112-------------- 8-------------- 30-------------- 74

23

1/H 2/M 8/L 0-----1-----2-----3-----4 0-----4-----3-----2-----1 0-----3-----4-----2-----1 ¦1 2 3 4 ¦0 ¦1 0 4 3 ¦2 ¦1 0 3 4 ¦2 1 2 2 3 3 4 4 0 0 1 1 2 0 1 4 0 3 4 2 3 3 2 4 1 2 0 1 3 0 4 ¦2 3 3 4 4 0 0 1 ¦1 2 ¦2 3 1 2 0 1 4 0 ¦3 4 ¦0 4 3 2 4 1 2 0 ¦1 3 2 3 4-3-4-0-4-0-1-0-1-2-1-2-3 2 3 4-1-2-3-0-1-2-4-0-1-3-4-0 4 1 3-2-0-4-1-3-2-0-4-1-3-2-0 ¦3 4¦ 4 0 0 1 1 2 ¦2 3¦ ¦3 4¦ 2 3 1 2 0 1 ¦4 0¦ ¦3 2¦ 4 1 2 0 1 3 ¦0 4¦ 3 4 0 4 0 1 0 1 2 1 2 3 2 3 4 3 4 0 2 3 4 1 2 3 0 1 2 4 0 1 2 0 4 1 3 2 0 4 1 3 2 0 4 1 3 ¦4 0¦ 0 1 1 2 2 3 ¦3 4¦ ¦4 0¦ 3 4 2 3 1 2 ¦0 1¦ ¦4 1¦ 2 0 1 3 0 4 ¦3 2¦ 4-0-1-0-1-2-1-2-3-2-3-4-3 4 0 4-0-1-3-4-0-2-3-4-1-2-3-0 1 2 1-3-2-0-4-1-3-2-0-4-1-3-2 0 4 0 1¦ 1 2 2 3 3 4 4 0¦ 0 1¦ 4 0 3 4 2 3 1 2¦ 2 0¦ 1 3 0 4 3 2 4 1¦ 1 2 2 3 3 4 4 0 0 1 1 2 0 1 4 0 3 4 2 3 4 1 2 0 1 3 0 4 3 2 2¦ 3 4 0 1¦ 2¦ 1 0 4 3¦ 3¦ 4 2 1 0¦ 3-----4-----0-----1-----2 3-----2-----1-----0-----4 0-----3-----4-----2-----1

1/H 2/M 8/L 4# 1-------------- 49-------------- 70-------------- 88--------------107 ¦ 32 51 99 120 ¦ 13 34 63 55 82 98 101 117 24 11 45 ¦ 61 95 84 113 105 7 23 26 ¦ 42 74 65 92 124-- 83---111--- 20--102--- 9--- 38-- 21--- 30--- 57-- 44--- 73--- 76 ¦ 94 123 ¦ 115 17 8 36 27 59 ¦ 71 80 ¦ 93 121 5 112 19 48 6 40 67 29 58 86 75 77 109 ¦125 2 ¦ 18 46 37 69 56 90 ¦ 79 108 ¦ 122--- 4--- 33-- 16--- 50--- 52-- 39--- 68--- 96-- 60--- 87---119-- 78 106 15 3 31 ¦ 47 54 66 100 89 118 110 12 ¦ 35 62 53 81 97 104 116 25 14 43 64 ¦ 85 103 22 41 ¦ 91--------------114-------------- 10-------------- 28-------------- 72 . . . . . Let me list it out in another compact style now, since I have got too many solutions to present here. I like to skip listing each set of Latin Units with only one exception in the list. I will also skip listing many solutions themselves, because I just want to show you the general image of the structure of solution set, in spite of only an outline. *** 'Complete Euler Cubes' of Order 5 for the Pan-triagonal Type: *** ** Made by Kanji Setsuda on June 12, 2010 with MacOSX & Xcode 3.1.2 **

1/H 2/M 3/L 1# 1/H 4/M 2/L 121#

1----------47----------68----------89---------110 1----------50----------69----------88---------107 ¦32 53 99 120 ¦11 ¦32 51 100 119 ¦13

35 63 51 84 97 105 118 21 14 42 47 63 66 82 90 101 109 25 3 44

¦61 94 82 115 103 6 24 27 ¦45 73 ¦53 94 97 113 116 7 15 26 ¦34 75 64 92 125--85-113--16-101-- 9--37--22--30--58--43--71--79 68 84 125--87-103--19-106--22--38-- 5--41--57--49--65--76

¦95 123 ¦ 111 19 7 40 28 56 ¦74 77 ¦ ¦99 115 ¦ 118 9 12 28 31 72 ¦55 91 ¦

93 121 4 114 17 50 10 38 66 26 59 87 72 80 108 89 105 16 108 24 40 2 43 59 46 62 78 70 81 122 124 2 ¦ 20 48 36 69 57 90 ¦78 106 ¦ 120 6 ¦ 14 30 33 74 52 93 ¦96 112 ¦

122-- 5--33--18--46--54--39--67-100--60--88-116--76 109 12 110--21--37-- 4--45--56--48--64--80--67--83-124--86 102 18

3 31 ¦ 49 52 70 98 86 119 107 15 ¦ 11 27 ¦ 35 71 54 95 98 114 117 8 ¦ 34 62 55 83 96 104 117 25 13 41 42 58 61 77 85 121 104 20 23 39

65 ¦ 81 102 23 44 ¦ 73 ¦ 92 111 10 29 ¦

91---------112---------- 8----------29----------75 79---------123----------17----------36----------60

1/H 6/M 3/L 241# 1/H 8/M 2/L 361#

1----------42----------73----------89---------110 1----------45----------74----------88---------107 ¦32 53 94 125 ¦11 ¦32 51 95 124 ¦13

35 63 51 84 92 105 123 16 14 47 42 63 71 82 90 101 109 20 3 49

¦61 99 82 115 103 6 19 27 ¦50 68 ¦53 99 92 113 121 7 15 26 ¦34 70 64 97 120--85-113--21-101-- 9--37--17--30--58--48--66--79 73 84 120--87-103--24-106--17--38-- 5--46--57--44--65--76

100 118 ¦ 111 24 7 40 28 56 ¦69 77 ¦ ¦94 115 ¦ 123 9 12 28 31 67 ¦55 96 ¦

98 116 4 114 22 45 10 38 71 26 59 87 67 80 108 89 105 21 108 19 40 2 48 59 41 62 78 75 81 117 119 2 ¦ 25 43 36 74 57 90 ¦78 106 ¦ 125 6 ¦ 14 30 33 69 52 98 ¦91 112 ¦

117-- 5--33--23--41--54--39--72--95--60--88-121--76 109 12 110--16--37-- 4--50--56--43--64--80--72--83-119--86 102 23

3 31 ¦ 44 52 75 93 86 124 107 15 ¦ 11 27 ¦ 35 66 54 100 93 114 122 8 ¦ 34 62 55 83 91 104 122 20 13 46 47 58 61 77 85 116 104 25 18 39

65 ¦ 81 102 18 49 ¦ 68 ¦ 97 111 10 29 ¦

96---------112---------- 8----------29----------70 79---------118----------22----------36----------60

24

1/H 10/M 3/L 481# 1/H 12/M 2/L 601#

1----------47----------63----------94---------110 1----------50----------64----------93---------107

¦32 53 99 115 ¦16 ¦32 51 100 114 ¦18

35 68 51 84 97 105 113 21 19 37 47 68 61 82 95 101 109 25 3 39 ¦66 89 82 120 103 6 24 27 ¦40 73 ¦53 89 97 118 111 7 20 26 ¦34 75

69 87 125--85-118--11-101-- 9--42--22--30--58--38--71--79 63 84 125--92-103--14-106--22--43-- 5--36--57--49--70--76

¦90 123 ¦ 116 14 7 45 28 56 ¦74 77 ¦ ¦99 120 ¦ 113 9 17 28 31 72 ¦55 86 ¦ 88 121 4 119 12 50 10 43 61 26 59 92 72 80 108 94 105 11 108 24 45 2 38 59 46 67 78 65 81 122

124 2 ¦ 15 48 41 64 57 95 ¦78 106 ¦ 115 6 ¦ 19 30 33 74 52 88 ¦96 117 ¦

122-- 5--33--13--46--54--44--62-100--60--93-111--76 109 17 110--21--42-- 4--40--56--48--69--80--62--83-124--91 102 13 3 31 ¦ 49 52 65 98 91 114 107 20 ¦ 16 27 ¦ 35 71 54 90 98 119 112 8 ¦

34 67 55 83 96 104 112 25 18 36 37 58 66 77 85 121 104 15 23 44

70 ¦ 81 102 23 39 ¦ 73 ¦ 87 116 10 29 ¦ 86---------117---------- 8----------29----------75 79---------123----------12----------41----------60

1/H 14/M 3/L 721# 1/H 16/M 2/L 841#

1----------37----------73----------94---------110 1----------40----------74----------93---------107

¦32 53 89 125 ¦16 ¦32 51 90 124 ¦18

35 68 51 84 87 105 123 11 19 47 37 68 71 82 95 101 109 15 3 49 ¦66 99 82 120 103 6 14 27 ¦50 63 ¦53 99 87 118 121 7 20 26 ¦34 65

69 97 115--85-118--21-101-- 9--42--12--30--58--48--61--79 73 84 115--92-103--24-106--12--43-- 5--46--57--39--70--76

100 113 ¦ 116 24 7 45 28 56 ¦64 77 ¦ ¦89 120 ¦ 123 9 17 28 31 62 ¦55 96 ¦ 98 111 4 119 22 40 10 43 71 26 59 92 62 80 108 94 105 21 108 14 45 2 48 59 36 67 78 75 81 112

114 2 ¦ 25 38 41 74 57 95 ¦78 106 ¦ 125 6 ¦ 19 30 33 64 52 98 ¦86 117 ¦

112-- 5--33--23--36--54--44--72--90--60--93-121--76 109 17 110--11--42-- 4--50--56--38--69--80--72--83-114--91 102 23 3 31 ¦ 39 52 75 88 91 124 107 20 ¦ 16 27 ¦ 35 61 54 100 88 119 122 8 ¦

34 67 55 83 86 104 122 15 18 46 47 58 66 77 85 111 104 25 13 44

70 ¦ 81 102 13 49 ¦ 63 ¦ 97 116 10 29 ¦ 96---------117---------- 8----------29----------65 79---------113----------22----------41----------60

1/H 18/M 3/L 961# 1/H 20/M 2/L 1081#

1----------42----------63----------99---------110 1----------45----------64----------98---------107

¦32 53 94 115 ¦21 ¦32 51 95 114 ¦23 35 73 51 84 92 105 113 16 24 37 42 73 61 82 100 101 109 20 3 39

¦71 89 82 125 103 6 19 27 ¦40 68 ¦53 89 92 123 111 7 25 26 ¦34 70

74 87 120--85-123--11-101-- 9--47--17--30--58--38--66--79 63 84 120--97-103--14-106--17--48-- 5--36--57--44--75--76 ¦90 118 ¦ 121 14 7 50 28 56 ¦69 77 ¦ ¦94 125 ¦ 113 9 22 28 31 67 ¦55 86 ¦

88 116 4 124 12 45 10 48 61 26 59 97 67 80 108 99 105 11 108 19 50 2 38 59 41 72 78 65 81 117

119 2 ¦ 15 43 46 64 57 100 ¦78 106 ¦ 115 6 ¦ 24 30 33 69 52 88 ¦91 122 ¦ 117-- 5--33--13--41--54--49--62--95--60--98-111--76 109 22 110--16--47-- 4--40--56--43--74--80--62--83-119--96 102 13

3 31 ¦ 44 52 65 93 96 114 107 25 ¦ 21 27 ¦ 35 66 54 90 93 124 112 8 ¦

34 72 55 83 91 104 112 20 23 36 37 58 71 77 85 116 104 15 18 49 75 ¦ 81 102 18 39 ¦ 68 ¦ 87 121 10 29 ¦

86---------122---------- 8----------29----------70 79---------118----------12----------46----------60

1/H 22/M 3/L 1201# 1/H 24/M 2/L 1321#

1----------37----------68----------99---------110 1----------40----------69----------98---------107

¦32 53 89 120 ¦21 ¦32 51 90 119 ¦23 35 73 51 84 87 105 118 11 24 42 37 73 66 82 100 101 109 15 3 44

¦71 94 82 125 103 6 14 27 ¦45 63 ¦53 94 87 123 116 7 25 26 ¦34 65

74 92 115--85-123--16-101-- 9--47--12--30--58--43--61--79 68 84 115--97-103--19-106--12--48-- 5--41--57--39--75--76 ¦95 113 ¦ 121 19 7 50 28 56 ¦64 77 ¦ ¦89 125 ¦ 118 9 22 28 31 62 ¦55 91 ¦

93 111 4 124 17 40 10 48 66 26 59 97 62 80 108 99 105 16 108 14 50 2 43 59 36 72 78 70 81 112

114 2 ¦ 20 38 46 69 57 100 ¦78 106 ¦ 120 6 ¦ 24 30 33 64 52 93 ¦86 122 ¦ 112-- 5--33--18--36--54--49--67--90--60--98-116--76 109 22 110--11--47-- 4--45--56--38--74--80--67--83-114--96 102 18

3 31 ¦ 39 52 70 88 96 119 107 25 ¦ 21 27 ¦ 35 61 54 95 88 124 117 8 ¦

34 72 55 83 86 104 117 15 23 41 42 58 71 77 85 111 104 20 13 49 75 ¦ 81 102 13 44 ¦ 63 ¦ 92 121 10 29 ¦

91---------122---------- 8----------29----------65 79---------113----------17----------46----------60

1/H 26/M 3/L 1441# 1/H 28/M 2/L 1561#

1----------47----------68----------84---------115 1----------50----------69----------83---------112 ¦37 53 99 120 ¦ 6 ¦37 51 100 119 ¦ 8

40 58 51 89 97 105 118 21 9 42 47 58 66 87 85 101 114 25 3 44

¦56 94 87 110 103 11 24 27 ¦45 73 ¦53 94 97 108 116 12 10 26 ¦39 75 59 92 125--90-108--16-101--14--32--22--30--63--43--71--79 68 89 125--82-103--19-111--22--33-- 5--41--62--49--60--76

¦95 123 ¦ 106 19 12 35 28 61 ¦74 77 ¦ ¦99 110 ¦ 118 14 7 28 36 72 ¦55 91 ¦

93 121 4 109 17 50 15 33 66 26 64 82 72 80 113 84 105 16 113 24 35 2 43 64 46 57 78 70 86 122 124 2 ¦ 20 48 31 69 62 85 ¦78 111 ¦ 120 11 ¦ 9 30 38 74 52 93 ¦96 107 ¦

122-- 5--38--18--46--54--34--67-100--65--83-116--76 114 7 115--21--32-- 4--45--61--48--59--80--67--88-124--81 102 18

3 36 ¦ 49 52 70 98 81 119 112 10 ¦ 6 27 ¦ 40 71 54 95 98 109 117 13 ¦ 39 57 55 88 96 104 117 25 8 41 42 63 56 77 90 121 104 20 23 34

60 ¦ 86 102 23 44 ¦ 73 ¦ 92 106 15 29 ¦

91---------107----------13----------29----------75 79---------123----------17----------31----------65

25

1/H 30/M 3/L 1681# 1/H 32/M 2/L 1801#

1----------42----------73----------84---------115 1----------45----------74----------83---------112

¦37 53 94 125 ¦ 6 ¦37 51 95 124 ¦ 8

40 58 51 89 92 105 123 16 9 47 42 58 71 87 85 101 114 20 3 49 ¦56 99 87 110 103 11 19 27 ¦50 68 ¦53 99 92 108 121 12 10 26 ¦39 70

59 97 120--90-108--21-101--14--32--17--30--63--48--66--79 73 89 120--82-103--24-111--17--33-- 5--46--62--44--60--76

100 118 ¦ 106 24 12 35 28 61 ¦69 77 ¦ ¦94 110 ¦ 123 14 7 28 36 67 ¦55 96 ¦ 98 116 4 109 22 45 15 33 71 26 64 82 67 80 113 84 105 21 113 19 35 2 48 64 41 57 78 75 86 117

119 2 ¦ 25 43 31 74 62 85 ¦78 111 ¦ 125 11 ¦ 9 30 38 69 52 98 ¦91 107 ¦

117-- 5--38--23--41--54--34--72--95--65--83-121--76 114 7 115--16--32-- 4--50--61--43--59--80--72--88-119--81 102 23 3 36 ¦ 44 52 75 93 81 124 112 10 ¦ 6 27 ¦ 40 66 54 100 93 109 122 13 ¦

39 57 55 88 91 104 122 20 8 46 47 63 56 77 90 116 104 25 18 34

60 ¦ 86 102 18 49 ¦ 68 ¦ 97 106 15 29 ¦ 96---------107----------13----------29----------70 79---------118----------22----------31----------65

1/H 34/M 3/L 1921# 1/H 36/M 2/L 2041#

1----------47----------58----------94---------115 1----------50----------59----------93---------112

¦37 53 99 110 ¦16 ¦37 51 100 109 ¦18

40 68 51 89 97 105 108 21 19 32 47 68 56 87 95 101 114 25 3 34 ¦66 84 87 120 103 11 24 27 ¦35 73 ¦53 84 97 118 106 12 20 26 ¦39 75

69 82 125--90-118-- 6-101--14--42--22--30--63--33--71--79 58 89 125--92-103-- 9-111--22--43-- 5--31--62--49--70--76

¦85 123 ¦ 116 9 12 45 28 61 ¦74 77 ¦ ¦99 120 ¦ 108 14 17 28 36 72 ¦55 81 ¦ 83 121 4 119 7 50 15 43 56 26 64 92 72 80 113 94 105 6 113 24 45 2 33 64 46 67 78 60 86 122

124 2 ¦ 10 48 41 59 62 95 ¦78 111 ¦ 110 11 ¦ 19 30 38 74 52 83 ¦96 117 ¦

122-- 5--38-- 8--46--54--44--57-100--65--93-106--76 114 17 115--21--42-- 4--35--61--48--69--80--57--88-124--91 102 8 3 36 ¦ 49 52 60 98 91 109 112 20 ¦ 16 27 ¦ 40 71 54 85 98 119 107 13 ¦

39 67 55 88 96 104 107 25 18 31 32 63 66 77 90 121 104 10 23 44

70 ¦ 86 102 23 34 ¦ 73 ¦ 82 116 15 29 ¦ 81---------117----------13----------29----------75 79---------123---------- 7----------41----------65

1/H 42/M 3/L 2161# 1/H 44/M 2/L 2281#

1----------42----------58----------99---------115 1----------45----------59----------98---------112

¦37 53 94 110 ¦21 ¦37 51 95 109 ¦23 40 73 51 89 92 105 108 16 24 32 42 73 56 87 100 101 114 20 3 34

¦71 84 87 125 103 11 19 27 ¦35 68 ¦53 84 92 123 106 12 25 26 ¦39 70

74 82 120--90-123-- 6-101--14--47--17--30--63--33--66--79 58 89 120--97-103-- 9-111--17--48-- 5--31--62--44--75--76 ¦85 118 ¦ 121 9 12 50 28 61 ¦69 77 ¦ ¦94 125 ¦ 108 14 22 28 36 67 ¦55 81 ¦

83 116 4 124 7 45 15 48 56 26 64 97 67 80 113 99 105 6 113 19 50 2 33 64 41 72 78 60 86 117

119 2 ¦ 10 43 46 59 62 100 ¦78 111 ¦ 110 11 ¦ 24 30 38 69 52 83 ¦91 122 ¦ 117-- 5--38-- 8--41--54--49--57--95--65--98-106--76 114 22 115--16--47-- 4--35--61--43--74--80--57--88-119--96 102 8

3 36 ¦ 44 52 60 93 96 109 112 25 ¦ 21 27 ¦ 40 66 54 85 93 124 107 13 ¦

39 72 55 88 91 104 107 20 23 31 32 63 71 77 90 116 104 10 18 49 75 ¦ 86 102 18 34 ¦ 68 ¦ 82 121 15 29 ¦

81---------122----------13----------29----------70 79---------118---------- 7----------46----------65

1/H 50/M 3/L 2401# 1/H 52/M 2/L 2521#

1----------47----------63----------84---------120 1----------50----------64----------83---------117

¦42 53 99 115 ¦ 6 ¦42 51 100 114 ¦ 8 45 58 51 94 97 105 113 21 9 37 47 58 61 92 85 101 119 25 3 39

¦56 89 92 110 103 16 24 27 ¦40 73 ¦53 89 97 108 111 17 10 26 ¦44 75

59 87 125--95-108--11-101--19--32--22--30--68--38--71--79 63 94 125--82-103--14-116--22--33-- 5--36--67--49--60--76 ¦90 123 ¦ 106 14 17 35 28 66 ¦74 77 ¦ ¦99 110 ¦ 113 19 7 28 41 72 ¦55 86 ¦

88 121 4 109 12 50 20 33 61 26 69 82 72 80 118 84 105 11 118 24 35 2 38 69 46 57 78 65 91 122

124 2 ¦ 15 48 31 64 67 85 ¦78 116 ¦ 115 16 ¦ 9 30 43 74 52 88 ¦96 107 ¦ 122-- 5--43--13--46--54--34--62-100--70--83-111--76 119 7 120--21--32-- 4--40--66--48--59--80--62--93-124--81 102 13

3 41 ¦ 49 52 65 98 81 114 117 10 ¦ 6 27 ¦ 45 71 54 90 98 109 112 18 ¦

44 57 55 93 96 104 112 25 8 36 37 68 56 77 95 121 104 15 23 34 60 ¦ 91 102 23 39 ¦ 73 ¦ 87 106 20 29 ¦

86---------107----------18----------29----------75 79---------123----------12----------31----------70

1/H 58/M 3/L 2641# 1/H 60/M 2/L 2761#

1----------47----------58----------89---------120 1----------50----------59----------88---------117 ¦42 53 99 110 ¦11 ¦42 51 100 109 ¦13

45 63 51 94 97 105 108 21 14 32 47 63 56 92 90 101 119 25 3 34

¦61 84 92 115 103 16 24 27 ¦35 73 ¦53 84 97 113 106 17 15 26 ¦44 75 64 82 125--95-113-- 6-101--19--37--22--30--68--33--71--79 58 94 125--87-103-- 9-116--22--38-- 5--31--67--49--65--76

¦85 123 ¦ 111 9 17 40 28 66 ¦74 77 ¦ ¦99 115 ¦ 108 19 12 28 41 72 ¦55 81 ¦

83 121 4 114 7 50 20 38 56 26 69 87 72 80 118 89 105 6 118 24 40 2 33 69 46 62 78 60 91 122 124 2 ¦ 10 48 36 59 67 90 ¦78 116 ¦ 110 16 ¦ 14 30 43 74 52 83 ¦96 112 ¦

122-- 5--43-- 8--46--54--39--57-100--70--88-106--76 119 12 120--21--37-- 4--35--66--48--64--80--57--93-124--86 102 8

3 41 ¦ 49 52 60 98 86 109 117 15 ¦ 11 27 ¦ 45 71 54 85 98 114 107 18 ¦ 44 62 55 93 96 104 107 25 13 31 32 68 61 77 95 121 104 10 23 39

65 ¦ 91 102 23 34 ¦ 73 ¦ 82 111 20 29 ¦

81---------112----------18----------29----------75 79---------123---------- 7----------36----------70

26

1/H 98/M 3/L 2881# 1/H 100/M 2/L 3001#

6----------47----------68----------89---------105 6----------50----------69----------88---------102

¦27 58 99 120 ¦11 ¦27 56 100 119 ¦13

30 63 56 79 97 110 118 21 14 42 47 63 66 77 90 106 104 25 8 44 ¦61 94 77 115 108 1 24 32 ¦45 73 ¦58 94 97 113 116 2 15 31 ¦29 75

64 92 125--80-113--16-106-- 4--37--22--35--53--43--71--84 68 79 125--87-108--19-101--22--38--10--41--52--49--65--81

¦95 123 ¦ 111 19 2 40 33 51 ¦74 82 ¦ ¦99 115 ¦ 118 4 12 33 26 72 ¦60 91 ¦ 93 121 9 114 17 50 5 38 66 31 54 87 72 85 103 89 110 16 103 24 40 7 43 54 46 62 83 70 76 122

124 7 ¦ 20 48 36 69 52 90 ¦83 101 ¦ 120 1 ¦ 14 35 28 74 57 93 ¦96 112 ¦

122--10--28--18--46--59--39--67-100--55--88-116--81 104 12 105--21--37-- 9--45--51--48--64--85--67--78-124--86 107 18 8 26 ¦ 49 57 70 98 86 119 102 15 ¦ 11 32 ¦ 30 71 59 95 98 114 117 3 ¦

29 62 60 78 96 109 117 25 13 41 42 53 61 82 80 121 109 20 23 39

65 ¦ 76 107 23 44 ¦ 73 ¦ 92 111 5 34 ¦ 91---------112---------- 3----------34----------75 84---------123----------17----------36----------55

2/H 1/M 3/L

0-----4-----3-----2-----1 0-----1-----2-----3-----4 0-----1-----2-----3-----4

¦1 0 4 3 ¦2 ¦1 2 3 4 ¦0 ¦1 2 3 4 ¦0

1 2 0 1 4 0 3 4 2 3 1 2 2 3 3 4 4 0 0 1 4 2 0 3 1 4 2 0 3 1 ¦2 3 1 2 0 1 4 0 ¦3 4 ¦2 3 3 4 4 0 0 1 ¦1 2 ¦0 3 1 4 2 0 3 1 ¦4 2

2 3 4-1-2-3-0-1-2-4-0-1-3-4-0 2 3 4-3-4-0-4-0-1-0-1-2-1-2-3 3 1 4-4-2-0-0-3-1-1-4-2-2-0-3

¦3 4¦ 2 3 1 2 0 1 ¦4 0¦ ¦3 4¦ 4 0 0 1 1 2 ¦2 3¦ ¦4 2¦ 0 3 1 4 2 0 ¦3 1¦ 3 4 0 2 3 4 1 2 3 0 1 2 4 0 1 3 4 0 4 0 1 0 1 2 1 2 3 2 3 4 2 0 3 3 1 4 4 2 0 0 3 1 1 4 2

¦4 0¦ 3 4 2 3 1 2 ¦0 1¦ ¦4 0¦ 0 1 1 2 2 3 ¦3 4¦ ¦3 1¦ 4 2 0 3 1 4 ¦2 0¦

4-0-1-3-4-0-2-3-4-1-2-3-0 1 2 4-0-1-0-1-2-1-2-3-2-3-4-3 4 0 1-4-2-2-0-3-3-1-4-4-2-0-0 3 1 0 1¦ 4 0 3 4 2 3 1 2¦ 0 1¦ 1 2 2 3 3 4 4 0¦ 2 0¦ 3 1 4 2 0 3 1 4¦

1 2 0 1 4 0 3 4 2 3 1 2 2 3 3 4 4 0 0 1 3 1 4 2 0 3 1 4 2 0

2¦ 1 0 4 3¦ 2¦ 3 4 0 1¦ 4¦ 0 1 2 3¦ 3-----2-----1-----0-----4 3-----4-----0-----1-----2 0-----1-----2-----3-----4

2/H 1/M 3/L 14401#

1--------------107-------------- 88-------------- 69-------------- 50

¦ 32 13 119 100 ¦ 51 35 63 11 44 117 25 98 101 54 82

¦ 61 94 42 75 23 26 104 7 ¦ 85 113

64 92 125-- 45--- 73--- 76-- 21--- 29--- 57--102--- 10--- 38-- 83---111--- 19 ¦ 95 123 ¦ 71 79 27 60 8 36 ¦114 17 ¦

93 121 4 74 77 110 30 58 86 6 39 67 112 20 48

¦124 2 ¦ 80 108 56 89 37 70 ¦ 18 46 ¦ 122--- 5--- 33-- 78---106--- 14-- 59--- 87---120-- 40--- 68--- 96-- 16 49 52

3 31 ¦ 109 12 90 118 66 99 47 55 ¦

34 62 15 43 116 24 97 105 53 81 65 ¦ 41 22 103 84 ¦

91-------------- 72-------------- 28-------------- 9--------------115

4/H 1/M 2/L 28801# 5/H 2/M 3/L 43201#

1---------110----------89----------68----------47 1----------47----------68---------114----------85 ¦32 11 120 99 ¦53 ¦32 53 124 95 ¦11

107 63 86 42 70 21 49 105 3 84 35 63 51 109 122 80 93 21 14 42

¦13 94 117 73 96 27 55 6 ¦34 115 ¦61 119 107 90 78 6 24 27 ¦45 73 88 44 125--67--23--79--46-102--58-- 5--81--37-109--65--16 64 117 100-110--88--16--76-- 9--37--22--30--58--43--71-104

119 75 ¦ 98 29 52 8 31 112 ¦15 91 ¦ 120 98 ¦ 86 19 7 40 28 56 ¦74 102 ¦

69 25 76 48 104 60 2 83 39 106 62 18 90 41 122 118 96 4 89 17 50 10 38 66 26 59 112 72 105 83 100 26 ¦ 54 10 33 114 12 93 116 72 ¦ ¦99 2 ¦ 20 48 36 69 57 115 103 81 ¦

50-101--57-- 4--85--36-108--64--20--87--43-124--66 22 78 97-- 5--33--18--46--54--39--67-125--60-113--91-101 84 12

51 7 ¦ 35 111 14 95 118 74 97 28 ¦ 3 31 ¦ 49 52 70 123 111 94 82 15 ¦ 82 38 61 17 45 121 24 80 103 59 34 62 55 108 121 79 92 25 13 41

113 ¦ 92 71 30 9 ¦ 65 ¦ 106 77 23 44 ¦

19---------123----------77----------56----------40 116----------87---------- 8----------29----------75

6/H 1/M 3/L 57601# 8/H 1/M 2/L 72001#

1----------82---------113----------69----------50 1----------85---------114----------68----------47 ¦32 13 94 125 ¦51 ¦32 11 95 124 ¦53

35 63 11 44 92 25 123 76 54 107 82 63 111 42 70 21 49 80 3 109

¦61 119 42 75 23 26 79 7 110 88 ¦13 119 92 73 121 27 55 6 ¦34 90 64 117 100--45--73-101--21--29--57--77--10--38-108--86--19 113 44 100--67--23-104--46--77--58-- 5-106--37--84--65--16

120 98 ¦ 71 104 27 60 8 36 ¦89 17 ¦ ¦94 75 ¦ 123 29 52 8 31 87 ¦15 116 ¦

118 96 4 74 102 85 30 58 111 6 39 67 87 20 48 69 25 101 48 79 60 2 108 39 81 62 18 115 41 97 ¦99 2 ¦ 105 83 56 114 37 70 ¦18 46 ¦ 125 26 ¦ 54 10 33 89 12 118 ¦91 72 ¦

97-- 5--33-103--81--14--59-112--95--40--68-121--16 49 52 50--76--57-- 4-110--36--83--64--20-112--43--99--66 22 103

3 31 ¦ 84 12 115 93 66 124 47 55 ¦ 51 7 ¦ 35 86 14 120 93 74 122 28 ¦ 34 62 15 43 91 24 122 80 53 106 107 38 61 17 45 96 24 105 78 59

65 ¦ 41 22 78 109 ¦ 88 ¦ 117 71 30 9 ¦

116----------72----------28---------- 9----------90 19----------98---------102----------56----------40

27

9/H 2/M 3/L 86401# 10/H 1/M 3/L 100801#

1----------47----------93----------64---------110 1---------107----------63----------94----------50

¦32 78 74 120 ¦11 ¦32 13 119 75 ¦76

35 88 76 59 72 105 118 21 14 42 35 88 11 44 117 25 73 101 79 57 ¦86 69 57 115 103 6 24 27 ¦45 98 ¦86 69 42 100 23 26 104 7 ¦60 113

89 67 125--60-113--16-101-- 9--37--22--30--83--43--96--54 89 67 125--45--98--51--21--29--82-102--10--38--58-111--19

¦70 123 ¦ 111 19 7 40 28 81 ¦99 52 ¦ ¦70 123 ¦ 96 54 27 85 8 36 ¦114 17 ¦ 68 121 4 114 17 50 10 38 91 26 84 62 97 55 108 68 121 4 99 52 110 30 83 61 6 39 92 112 20 48

124 2 ¦ 20 48 36 94 82 65 ¦53 106 ¦ 124 2 ¦ 55 108 81 64 37 95 ¦18 46 ¦

122-- 5--33--18--46--79--39--92--75--85--63-116--51 109 12 122-- 5--33--53-106--14--84--62-120--40--93--71--16 49 77 3 31 ¦ 49 77 95 73 61 119 107 15 ¦ 3 31 ¦ 109 12 65 118 91 74 47 80 ¦

34 87 80 58 71 104 117 25 13 41 34 87 15 43 116 24 72 105 78 56

90 ¦ 56 102 23 44 ¦ 90 ¦ 41 22 103 59 ¦ 66---------112---------- 8----------29---------100 66----------97----------28---------- 9---------115

12/H 1/M 2/L 115201# 13/H 2/M 3/L 129601#

1---------110----------64----------93----------47 1----------47----------93---------114----------60

¦32 11 120 74 ¦78 ¦32 78 124 70 ¦11

107 88 61 42 95 21 49 105 3 59 35 88 76 109 122 55 68 21 14 42 ¦13 69 117 98 71 27 80 6 ¦34 115 ¦86 119 107 65 53 6 24 27 ¦45 98

63 44 125--92--23--54--46-102--83-- 5--56--37-109--90--16 89 117 75-110--63--16--51-- 9--37--22--30--83--43--96-104

119 100 ¦ 73 29 77 8 31 112 ¦15 66 ¦ 120 73 ¦ 61 19 7 40 28 81 ¦99 102 ¦ 94 25 51 48 104 85 2 58 39 106 87 18 65 41 122 118 71 4 64 17 50 10 38 91 26 84 112 97 105 58

¦75 26 ¦ 79 10 33 114 12 68 116 97 ¦ ¦74 2 ¦ 20 48 36 94 82 115 103 56 ¦

50-101--82-- 4--60--36-108--89--20--62--43-124--91 22 53 72-- 5--33--18--46--79--39--92-125--85-113--66-101 59 12 76 7 ¦ 35 111 14 70 118 99 72 28 ¦ 3 31 ¦ 49 77 95 123 111 69 57 15 ¦

57 38 86 17 45 121 24 55 103 84 34 87 80 108 121 54 67 25 13 41

113 ¦ 67 96 30 9 ¦ 90 ¦ 106 52 23 44 ¦ 19---------123----------52----------81----------40 116----------62---------- 8----------29---------100

14/H 1/M 3/L 144001# 16/H 1/M 2/L 158401#

1----------57---------113----------94----------50 1----------60---------114----------93----------47

¦32 13 69 125 ¦76 ¦32 11 70 124 ¦78 35 88 11 44 67 25 123 51 79 107 57 88 111 42 95 21 49 55 3 109

¦86 119 42 100 23 26 54 7 110 63 ¦13 119 67 98 121 27 80 6 ¦34 65

89 117 75--45--98-101--21--29--82--52--10--38-108--61--19 113 44 75--92--23-104--46--52--83-- 5-106--37--59--90--16 120 73 ¦ 96 104 27 85 8 36 ¦64 17 ¦ ¦69 100 ¦ 123 29 77 8 31 62 ¦15 116 ¦

118 71 4 99 102 60 30 83 111 6 39 92 62 20 48 94 25 101 48 54 85 2 108 39 56 87 18 115 41 72

¦74 2 ¦ 105 58 81 114 37 95 ¦18 46 ¦ 125 26 ¦ 79 10 33 64 12 118 ¦66 97 ¦ 72-- 5--33-103--56--14--84-112--70--40--93-121--16 49 77 50--51--82-- 4-110--36--58--89--20-112--43--74--91 22 103

3 31 ¦ 59 12 115 68 91 124 47 80 ¦ 76 7 ¦ 35 61 14 120 68 99 122 28 ¦

34 87 15 43 66 24 122 55 78 106 107 38 86 17 45 71 24 105 53 84 90 ¦ 41 22 53 109 ¦ 63 ¦ 117 96 30 9 ¦

116----------97----------28---------- 9----------65 19----------73---------102----------81----------40

17/H 2/M 3/L 172801# 18/H 1/M 3/L 187201#

1----------47---------118----------64----------85 1----------82----------63---------119----------50

¦32 103 74 95 ¦11 ¦32 13 94 75 ¦101 35 113 101 59 72 80 93 21 14 42 35 113 11 44 92 25 73 76 104 57

111 69 57 90 78 6 24 27 ¦45 123 111 69 42 125 23 26 79 7 ¦60 88

114 67 100--60--88--16--76-- 9--37--22--30-108--43-121--54 114 67 100--45-123--51--21--29-107--77--10--38--58--86--19 ¦70 98 ¦ 86 19 7 40 28 106 124 52 ¦ ¦70 98 ¦ 121 54 27 110 8 36 ¦89 17 ¦

68 96 4 89 17 50 10 38 116 26 109 62 122 55 83 68 96 4 124 52 85 30 108 61 6 39 117 87 20 48

¦99 2 ¦ 20 48 36 119 107 65 ¦53 81 ¦ ¦99 2 ¦ 55 83 106 64 37 120 ¦18 46 ¦ 97-- 5--33--18--46-104--39-117--75-110--63--91--51 84 12 97-- 5--33--53--81--14-109--62--95--40-118--71--16 49 102

3 31 ¦ 49 102 120 73 61 94 82 15 ¦ 3 31 ¦ 84 12 65 93 116 74 47 105 ¦

34 112 105 58 71 79 92 25 13 41 34 112 15 43 91 24 72 80 103 56 115 ¦ 56 77 23 44 ¦ 115 ¦ 41 22 78 59 ¦

66----------87---------- 8----------29---------125 66---------122----------28---------- 9----------90

20/H 1/M 2/L 201601# 21/H 2/M 3/L 216001#

1----------85----------64---------118----------47 1----------47---------118----------89----------60 ¦32 11 95 74 103 ¦32 103 99 70 ¦11

82 113 61 42 120 21 49 80 3 59 35 113 101 84 97 55 68 21 14 42

¦13 69 92 123 71 27 105 6 ¦34 90 111 94 82 65 53 6 24 27 ¦45 123 63 44 100-117--23--54--46--77-108-- 5--56--37--84-115--16 114 92 75--85--63--16--51-- 9--37--22--30-108--43--21--79

¦94 125 ¦ 73 29 102 8 31 87 ¦15 66 ¦ ¦95 73 ¦ 61 19 7 40 28 106 124 77 ¦

119 25 51 48 79 110 2 58 39 81 112 18 65 41 97 93 71 4 64 17 50 10 38 116 26 109 87 122 80 58 ¦75 26 ¦ 104 10 33 89 12 68 ¦91 122 ¦ ¦74 2 ¦ 20 48 36 119 107 90 ¦78 56 ¦

50--76-107-- 4--60--36--83-114--20--62--43--99-116 22 53 72-- 5--33--18--46-104--39-117-100-110--88--66--76 59 12

101 7 ¦ 35 86 14 70 93 124 72 28 ¦ 3 31 ¦ 49 102 120 98 86 69 57 15 ¦ 57 38 111 17 45 96 24 55 78 109 34 112 105 83 96 54 67 25 13 41

88 ¦ 67 121 30 9 ¦ 115 ¦ 81 52 23 44 ¦

19----------98----------52---------106----------40 91----------62---------- 8----------29---------125

28

22/H 1/M 3/L 230401# 24/H 1/M 2/L 244801#

1----------57----------88---------119----------50 1----------60----------89---------118----------47

¦32 13 69 100 101 ¦32 11 70 99 103

35 113 11 44 67 25 98 51 104 82 57 113 86 42 120 21 49 55 3 84 111 94 42 125 23 26 54 7 ¦85 63 ¦13 94 67 123 96 27 105 6 ¦34 65

114 92 75--45-123--76--21--29-107--52--10--38--83--61--19 88 44 75-117--23--79--46--52-108-- 5--81--37--59-115--16

¦95 73 ¦ 121 79 27 110 8 36 ¦64 17 ¦ ¦69 125 ¦ 98 29 102 8 31 62 ¦15 91 ¦ 93 71 4 124 77 60 30 108 86 6 39 117 62 20 48 119 25 76 48 54 110 2 83 39 56 112 18 90 41 72

¦74 2 ¦ 80 58 106 89 37 120 ¦18 46 ¦ 100 26 ¦ 104 10 33 64 12 93 ¦66 122 ¦

72-- 5--33--78--56--14-109--87--70--40-118--96--16 49 102 50--51-107-- 4--85--36--58-114--20--87--43--74-116 22 78 3 31 ¦ 59 12 90 68 116 99 47 105 ¦ 101 7 ¦ 35 61 14 95 68 124 97 28 ¦

34 112 15 43 66 24 97 55 103 81 82 38 111 17 45 71 24 80 53 109

115 ¦ 41 22 53 84 ¦ 63 ¦ 92 121 30 9 ¦ 91---------122----------28---------- 9----------65 19----------73----------77---------106----------40

25/H 2/M 3/L 259201# 26/H 1/M 3/L 273601#

1----------72----------43----------89---------110 1---------107----------88----------44----------75

¦57 28 99 120 ¦11 ¦57 13 119 100 ¦26

60 38 26 84 97 105 118 21 14 67 60 38 11 69 117 25 98 101 29 82 ¦36 94 82 115 103 6 24 52 ¦70 48 ¦36 94 67 50 23 51 104 7 ¦85 113

39 92 125--85-113--16-101-- 9--62--22--55--33--68--46--79 39 92 125--70--48--76--21--54--32-102--10--63--83-111--19

¦95 123 ¦ 111 19 7 65 53 31 ¦49 77 ¦ ¦95 123 ¦ 46 79 52 35 8 61 114 17 ¦ 93 121 4 114 17 75 10 63 41 51 34 87 47 80 108 93 121 4 49 77 110 55 33 86 6 64 42 112 20 73

124 2 ¦ 20 73 61 44 32 90 ¦78 106 ¦ 124 2 ¦ 80 108 31 89 62 45 ¦18 71 ¦

122-- 5--58--18--71--29--64--42-100--35--88-116--76 109 12 122-- 5--58--78-106--14--34--87-120--65--43--96--16 74 27 3 56 ¦ 74 27 45 98 86 119 107 15 ¦ 3 56 ¦ 109 12 90 118 41 99 72 30 ¦

59 37 30 83 96 104 117 25 13 66 59 37 15 68 116 24 97 105 28 81

40 ¦ 81 102 23 69 ¦ 40 ¦ 66 22 103 84 ¦ 91---------112---------- 8----------54----------50 91----------47----------53---------- 9---------115

28/H 1/M 2/L 288001# 29/H 2/M 3/L 302401#

1---------110----------89----------43----------72 1----------72----------43---------114----------85

¦57 11 120 99 ¦28 ¦57 28 124 95 ¦11 107 38 86 67 45 21 74 105 3 84 60 38 26 109 122 80 93 21 14 67

¦13 94 117 48 96 52 30 6 ¦59 115 ¦36 119 107 90 78 6 24 52 ¦70 48

88 69 125--42--23--79--71-102--33-- 5--81--62-109--40--16 39 117 100-110--88--16--76-- 9--62--22--55--33--68--46-104 119 50 ¦ 98 54 27 8 56 112 ¦15 91 ¦ 120 98 ¦ 86 19 7 65 53 31 ¦49 102 ¦

44 25 76 73 104 35 2 83 64 106 37 18 90 66 122 118 96 4 89 17 75 10 63 41 51 34 112 47 105 83

100 51 ¦ 29 10 58 114 12 93 116 47 ¦ ¦99 2 ¦ 20 73 61 44 32 115 103 81 ¦ 75-101--32-- 4--85--61-108--39--20--87--68-124--41 22 78 97-- 5--58--18--71--29--64--42-125--35-113--91-101 84 12

26 7 ¦ 60 111 14 95 118 49 97 53 ¦ 3 56 ¦ 74 27 45 123 111 94 82 15 ¦

82 63 36 17 70 121 24 80 103 34 59 37 30 108 121 79 92 25 13 66 113 ¦ 92 46 55 9 ¦ 40 ¦ 106 77 23 69 ¦

19---------123----------77----------31----------65 116----------87---------- 8----------54----------50

30/H 1/M 3/L 316801# 32/H 1/M 2/L 331201#

1----------82---------113----------44----------75 1----------85---------114----------43----------72

¦57 13 94 125 ¦26 ¦57 11 95 124 ¦28 60 38 11 69 92 25 123 76 29 107 82 38 111 67 45 21 74 80 3 109

¦36 119 67 50 23 51 79 7 110 88 ¦13 119 92 48 121 52 30 6 ¦59 90

39 117 100--70--48-101--21--54--32--77--10--63-108--86--19 113 69 100--42--23-104--71--77--33-- 5-106--62--84--40--16 120 98 ¦ 46 104 52 35 8 61 ¦89 17 ¦ ¦94 50 ¦ 123 54 27 8 56 87 ¦15 116 ¦

118 96 4 49 102 85 55 33 111 6 64 42 87 20 73 44 25 101 73 79 35 2 108 64 81 37 18 115 66 97

¦99 2 ¦ 105 83 31 114 62 45 ¦18 71 ¦ 125 51 ¦ 29 10 58 89 12 118 ¦91 47 ¦ 97-- 5--58-103--81--14--34-112--95--65--43-121--16 74 27 75--76--32-- 4-110--61--83--39--20-112--68--99--41 22 103

3 56 ¦ 84 12 115 93 41 124 72 30 ¦ 26 7 ¦ 60 86 14 120 93 49 122 53 ¦

59 37 15 68 91 24 122 80 28 106 107 63 36 17 70 96 24 105 78 34 40 ¦ 66 22 78 109 ¦ 88 ¦ 117 46 55 9 ¦

116----------47----------53---------- 9----------90 19----------98---------102----------31----------65

33/H 2/M 3/L 345601# 34/H 1/M 3/L 360001#

1----------72----------93----------39---------110 1---------107----------38----------94----------75 ¦57 78 49 120 ¦11 ¦57 13 119 50 ¦76

60 88 76 34 47 105 118 21 14 67 60 88 11 69 117 25 48 101 79 32

¦86 44 32 115 103 6 24 52 ¦70 98 ¦86 44 67 100 23 51 104 7 ¦35 113 89 42 125--35-113--16-101-- 9--62--22--55--83--68--96--29 89 42 125--70--98--26--21--54--82-102--10--63--33-111--19

¦45 123 ¦ 111 19 7 65 53 81 ¦99 27 ¦ ¦45 123 ¦ 96 29 52 85 8 61 114 17 ¦

43 121 4 114 17 75 10 63 91 51 84 37 97 30 108 43 121 4 99 27 110 55 83 36 6 64 92 112 20 73 124 2 ¦ 20 73 61 94 82 40 ¦28 106 ¦ 124 2 ¦ 30 108 81 39 62 95 ¦18 71 ¦

122-- 5--58--18--71--79--64--92--50--85--38-116--26 109 12 122-- 5--58--28-106--14--84--37-120--65--93--46--16 74 77

3 56 ¦ 74 77 95 48 36 119 107 15 ¦ 3 56 ¦ 109 12 40 118 91 49 72 80 ¦ 59 87 80 33 46 104 117 25 13 66 59 87 15 68 116 24 47 105 78 31

90 ¦ 31 102 23 69 ¦ 90 ¦ 66 22 103 34 ¦

41---------112---------- 8----------54---------100 41----------97----------53---------- 9---------115

29

36/H 1/M 2/L 374401# 40/H 2/M 3/L 388801#

1---------110----------39----------93----------72 1----------72----------93---------114----------35

¦57 11 120 49 ¦78 ¦57 78 124 45 ¦11

107 88 36 67 95 21 74 105 3 34 60 88 76 109 122 30 43 21 14 67 ¦13 44 117 98 46 52 80 6 ¦59 115 ¦86 119 107 40 28 6 24 52 ¦70 98

38 69 125--92--23--29--71-102--83-- 5--31--62-109--90--16 89 117 50-110--38--16--26-- 9--62--22--55--83--68--96-104

119 100 ¦ 48 54 77 8 56 112 ¦15 41 ¦ 120 48 ¦ 36 19 7 65 53 81 ¦99 102 ¦ 94 25 26 73 104 85 2 33 64 106 87 18 40 66 122 118 46 4 39 17 75 10 63 91 51 84 112 97 105 33

¦50 51 ¦ 79 10 58 114 12 43 116 97 ¦ ¦49 2 ¦ 20 73 61 94 82 115 103 31 ¦

75-101--82-- 4--35--61-108--89--20--37--68-124--91 22 28 47-- 5--58--18--71--79--64--92-125--85-113--41-101 34 12 76 7 ¦ 60 111 14 45 118 99 47 53 ¦ 3 56 ¦ 74 77 95 123 111 44 32 15 ¦

32 63 86 17 70 121 24 30 103 84 59 87 80 108 121 29 42 25 13 66

113 ¦ 42 96 55 9 ¦ 90 ¦ 106 27 23 69 ¦ 19---------123----------27----------81----------65 116----------37---------- 8----------54---------100

41/H 2/M 3/L 403201# 42/H 1/M 3/L 417601#

1----------72---------118----------39----------85 1----------82----------38---------119----------75

¦57 103 49 95 ¦11 ¦57 13 94 50 101

60 113 101 34 47 80 93 21 14 67 60 113 11 69 92 25 48 76 104 32 111 44 32 90 78 6 24 52 ¦70 123 111 44 67 125 23 51 79 7 ¦35 88

114 42 100--35--88--16--76-- 9--62--22--55-108--68-121--29 114 42 100--70-123--26--21--54-107--77--10--63--33--86--19

¦45 98 ¦ 86 19 7 65 53 106 124 27 ¦ ¦45 98 ¦ 121 29 52 110 8 61 ¦89 17 ¦ 43 96 4 89 17 75 10 63 116 51 109 37 122 30 83 43 96 4 124 27 85 55 108 36 6 64 117 87 20 73

¦99 2 ¦ 20 73 61 119 107 40 ¦28 81 ¦ ¦99 2 ¦ 30 83 106 39 62 120 ¦18 71 ¦

97-- 5--58--18--71-104--64-117--50-110--38--91--26 84 12 97-- 5--58--28--81--14-109--37--95--65-118--46--16 74 102 3 56 ¦ 74 102 120 48 36 94 82 15 ¦ 3 56 ¦ 84 12 40 93 116 49 72 105 ¦

59 112 105 33 46 79 92 25 13 66 59 112 15 68 91 24 47 80 103 31

115 ¦ 31 77 23 69 ¦ 115 ¦ 66 22 78 34 ¦ 41----------87---------- 8----------54---------125 41---------122----------53---------- 9----------90

44/H 1/M 2/L 432001# 48/H 2/M 3/L 446401#

1----------85----------39---------118----------72 1----------72---------118----------89----------35

¦57 11 95 49 103 ¦57 103 99 45 ¦11 82 113 36 67 120 21 74 80 3 34 60 113 101 84 97 30 43 21 14 67

¦13 44 92 123 46 52 105 6 ¦59 90 111 94 82 40 28 6 24 52 ¦70 123

38 69 100-117--23--29--71--77-108-- 5--31--62--84-115--16 114 92 50--85--38--16--26-- 9--62--22--55-108--68-121--79 ¦94 125 ¦ 48 54 102 8 56 87 ¦15 41 ¦ ¦95 48 ¦ 36 19 7 65 53 106 124 77 ¦

119 25 26 73 79 110 2 33 64 81 112 18 40 66 97 93 46 4 39 17 75 10 63 116 51 109 87 122 80 33

¦50 51 ¦ 104 10 58 89 12 43 ¦91 122 ¦ ¦49 2 ¦ 20 73 61 119 107 90 ¦78 31 ¦ 75--76-107-- 4--35--61--83-114--20--37--68--99-116 22 28 47-- 5--58--18--71-104--64-117-100-110--88--41--76 34 12

101 7 ¦ 60 86 14 45 93 124 47 53 ¦ 3 56 ¦ 74 102 120 98 86 44 32 15 ¦

32 63 111 17 70 96 24 30 78 109 59 112 105 83 96 29 42 25 13 66 88 ¦ 42 121 55 9 ¦ 115 ¦ 81 27 23 69 ¦

19----------98----------27---------106----------65 91----------37---------- 8----------54---------125

122/H 1/M 3/L 777601# 126/H 1/M 3/L 792001#

26---------107----------88----------19----------75 26----------82---------113----------19----------75

¦57 38 119 100 ¦ 1 ¦57 38 94 125 ¦ 1 60 13 36 69 117 50 98 101 4 82 60 13 36 69 92 50 123 76 4 107

¦11 94 67 25 48 51 104 32 ¦85 113 ¦11 119 67 25 48 51 79 32 110 88

14 92 125--70--23--76--46--54-- 7-102--35--63--83-111--44 14 117 100--70--23-101--46--54-- 7--77--35--63-108--86--44 ¦95 123 ¦ 21 79 52 10 33 61 114 42 ¦ 120 98 ¦ 21 104 52 10 33 61 ¦89 42 ¦

93 121 29 24 77 110 55 8 86 31 64 17 112 45 73 118 96 29 24 102 85 55 8 111 31 64 17 87 45 73

124 27 ¦ 80 108 6 89 62 20 ¦43 71 ¦ ¦99 27 ¦ 105 83 6 114 62 20 ¦43 71 ¦ 122--30--58--78-106--39-- 9--87-120--65--18--96--41 74 2 97--30--58-103--81--39-- 9-112--95--65--18-121--41 74 2

28 56 ¦ 109 37 90 118 16 99 72 5 ¦ 28 56 ¦ 84 37 115 93 16 124 72 5 ¦

59 12 40 68 116 49 97 105 3 81 59 12 40 68 91 49 122 80 3 106 15 ¦ 66 47 103 84 ¦ 15 ¦ 66 47 78 109 ¦

91----------22----------53----------34---------115 116----------22----------53----------34----------90

132/H 1/M 2/L 806401# 140/H 1/M 2/L 820801#

26---------110----------14----------93----------72 26----------85----------14---------118----------72 ¦57 36 120 24 ¦78 ¦57 36 95 24 103

107 88 11 67 95 46 74 105 28 9 82 113 11 67 120 46 74 80 28 9

¦38 19 117 98 21 52 80 31 ¦59 115 ¦38 19 92 123 21 52 105 31 ¦59 90 13 69 125--92--48-- 4--71-102--83--30-- 6--62-109--90--41 13 69 100-117--48-- 4--71--77-108--30-- 6--62--84-115--41

119 100 ¦ 23 54 77 33 56 112 ¦40 16 ¦ ¦94 125 ¦ 23 54 102 33 56 87 ¦40 16 ¦

94 50 1 73 104 85 27 8 64 106 87 43 15 66 122 119 50 1 73 79 110 27 8 64 81 112 43 15 66 97 ¦25 51 ¦ 79 35 58 114 37 18 116 97 ¦ ¦25 51 ¦ 104 35 58 89 37 18 ¦91 122 ¦

75-101--82--29--10--61-108--89--45--12--68-124--91 47 3 75--76-107--29--10--61--83-114--45--12--68--99-116 47 3

76 32 ¦ 60 111 39 20 118 99 22 53 ¦ 101 32 ¦ 60 86 39 20 93 124 22 53 ¦ 7 63 86 42 70 121 49 5 103 84 7 63 111 42 70 96 49 5 78 109

113 ¦ 17 96 55 34 ¦ 88 ¦ 17 121 55 34 ¦

44---------123---------- 2----------81----------65 44----------98---------- 2---------106----------65

30

146/H 1/M 3/L 835201# 156/H 1/M 2/L 849601#

26---------107----------63----------19----------100 26---------110----------14----------68----------97

¦82 38 119 75 ¦ 1 ¦82 36 120 24 ¦53

85 13 36 94 117 50 73 101 4 57 107 63 11 92 70 46 99 105 28 9 ¦11 69 92 25 48 76 104 32 ¦60 113 ¦38 19 117 73 21 77 55 31 ¦84 115

14 67 125--95--23--51--46--79-- 7-102--35--88--58-111--44 13 94 125--67--48-- 4--96-102--58--30-- 6--87-109--65--41

¦70 123 ¦ 21 54 77 10 33 86 114 42 ¦ 119 75 ¦ 23 79 52 33 81 112 ¦40 16 ¦ 68 121 29 24 52 110 80 8 61 31 89 17 112 45 98 69 50 1 98 104 60 27 8 89 106 62 43 15 91 122

124 27 ¦ 55 108 6 64 87 20 ¦43 96 ¦ ¦25 76 ¦ 54 35 83 114 37 18 116 72 ¦

122--30--83--53-106--39-- 9--62-120--90--18--71--41 99 2 100-101--57--29--10--86-108--64--45--12--93-124--66 47 3 28 81 ¦ 109 37 65 118 16 74 97 5 ¦ 51 32 ¦ 85 111 39 20 118 74 22 78 ¦

84 12 40 93 116 49 72 105 3 56 7 88 61 42 95 121 49 5 103 59

15 ¦ 91 47 103 59 ¦ 113 ¦ 17 71 80 34 ¦ 66----------22----------78----------34---------115 44---------123---------- 2----------56----------90

** Counts according to the Values of n1: ** 1:55296, 2:48384, 3:41472, 4:34560, 5:27648, 6:44928, 7:39744, 8:34560, 9:29376, 10:24192, 11:34560, 12:31104, 13:27648, 14:24192, 15:20736, 16:24192, 17:22464, 18:20736, 19:19008, 20:17280, 21:13824, 22:13824, 23:13824, 24:13824, 25:13824, 26:13824, 27:12096, 28:10368, 29: 8640, 30: 6912, 31:11232, 32: 9936, 33: 8640, 34: 7344, 35: 6048, 36: 8640, 37: 7776, 38: 6912, 39: 6048, 40: 5184, 41: 6048, 42: 5616, 43: 5184, 44: 4752, 45: 4320, 46: 3456, 47: 3456, 48: 3456, 49: 3456, 50: 3456, 51: 0, 52: 0, 53: 0, 54: 0, 55: 0, 56: 0, 57: 0, 58: 0, 59: 0, 60: 0, 61: 0, 62: 0, 63: 0, 64: 0, . . . . .

[Total Count of Standard Solutions = 864000] OK!

10. What I think about the result: How happy I am that I have counted all the solutions and can report about them! The list of solution counts above tells us all of them are the multiples of 27(=33): 3456=128x27; 4320=160x27; 4752=176x27; 5184=192x27; 5616=208x27; 6048=224x27; ... The total count is finally divided as: 864000 = 32000 x 27 = 28 x 33 x 53; What does it mean by these factors? What structure do they imply about? The total solution count also means: 480 x 480 x 480 x(3/8)x(1/48) = 864000. 480 is the count of Latin Units, 3/8 indicates the ratio of correct solutions to all the possible combinations and 1/48 means the ratio of standard solutions to primitive. The next and last job of ours is to make 'Complete Euler Cubes' of Order 5 for the Simultaneous Type: Both Self-complementary and Pan-triagonal. (English Version written by Kanji Setsuda on July 4, 2010 with MacOSX and Xcode3)

E-mail Address: [email protected]

make 'complete euler cubes' of order 5 for the pan...

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